Factorization Algebras in Quantum Field Theory [1] 9781316678626

Table of contents :
Contents......Page 8
1 Introduction......Page 12
1.1 The Motivating Example of Quantum Mechanics......Page 14
1.3 Prefactorization Algebras in Quantum Field Theory......Page 19
1.4 Comparisons with Other Formalizations of Quantum Field Theory......Page 22
1.5 Overview of This Volume......Page 27
1.6 Acknowledgments......Page 29
PART I PREFACTORIZATION ALGEBRAS......Page 32
2 From Gaussian Measures to Factorization Algebras......Page 34
2.1 Gaussian Integrals in Finite Dimensions......Page 36
2.2 Divergence in Infinite Dimensions......Page 38
2.3 The Prefactorization Structure on Observables......Page 42
2.4 From Quantum to Classical......Page 45
2.5 Correlation Functions......Page 47
2.6 Further Results on Free Field Theories......Page 50
2.7 Interacting Theories......Page 51
3.1 Prefactorization Algebras......Page 55
3.2 Associative Algebras from Prefactorization Algebras on R......Page 62
3.3 Modules as Defects......Page 63
3.4 A Construction of the Universal Enveloping Algebra......Page 70
3.5 Some Functional Analysis......Page 73
3.6 The Factorization Envelope of a Sheaf of Lie Algebras......Page 84
3.7 Equivariant Prefactorization Algebras......Page 90
PART II FIRST EXAMPLES OF FIELD THEORIES AND THEIR OBSERVABLES......Page 98
4.1 The Divergence Complex of a Measure......Page 100
4.2 The Prefactorization Algebra of a Free Field Theory......Page 104
4.3 Quantum Mechanics and the Weyl Algebra......Page 117
4.4 Pushforward and Canonical Quantization......Page 123
4.5 Abelian Chern–Simons Theory......Page 126
4.6 Another Take on Quantizing Classical Observables......Page 135
4.7 Correlation Functions......Page 140
4.8 Translation-Invariant Prefactorization Algebras......Page 142
4.9 States and Vacua for Translation Invariant Theories......Page 150
5.1 Vertex Algebras and Holomorphic Prefactorization Algebras on C......Page 156
5.2 Holomorphically Translation-Invariant Prefactorization Algebras......Page 160
5.3 A General Method for Constructing Vertex Algebras......Page 168
5.4 The βγ System and Vertex Algebras......Page 182
5.5 Kac–Moody Algebras and Factorization Envelopes......Page 199
PART III FACTORIZATION ALGEBRAS......Page 216
6.1 Factorization Algebras......Page 218
6.2 Factorization Algebras in Quantum Field Theory......Page 226
6.3 Variant Definitions of Factorization Algebras......Page 227
6.4 Locally Constant Factorization Algebras......Page 231
6.5 Factorization Algebras from Cosheaves......Page 236
6.6 Factorization Algebras from Local Lie Algebras......Page 241
7.1 Pushing Forward Factorization Algebras......Page 243
7.3 Pulling Back Along an Open Immersion......Page 251
7.4 Descent Along a Torsor......Page 252
8.1 Some Examples of Computations......Page 254
8.2 Abelian Chern–Simons Theory and Quantum Groups......Page 260
Appendix A Background......Page 284
Appendix B Functional Analysis......Page 321
Appendix C Homological Algebra in Differentiable Vector Spaces......Page 362
Appendix D The Atiyah–Bott Lemma......Page 385
References......Page 388
Index......Page 394

Citation preview

Factorization Algebras in Quantum Field Theory Volume 1 Factorization algebras are local-to-global objects that play a role in classical and quantum field theory that is similar to the role of sheaves in geometry: they conveniently organize complicated information. Their local structure encompasses examples such as associative and vertex algebras; in these examples, their global structure encompasses Hochschild homology and conformal blocks. In this first volume, the authors develop the theory of factorization algebras in depth, but with a focus upon examples exhibiting their use in field theory, such as the recovery of a vertex algebra from a chiral conformal field theory and a quantum group from Abelian Chern–Simons theory. Expositions of the relevant background in homological algebra, sheaves, and functional analysis are also included, thus making this book ideal for researchers and graduates working at the interface between mathematics and physics. K E V I N C O S T E L L O is the Krembil Foundation William Rowan Hamilton Chair in Theoretical Physics at the Perimeter Institute in Waterloo, Ontario. O W E N G W I L L I A M is a postdoctoral fellow at the Max Planck Institute for Mathematics in Bonn.

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M. Cabanes and M. Enguehard Representation Theory of Finite Reductive Groups J. B. Garnett and D. E. Marshall Harmonic Measure P. Cohn Free Ideal Rings and Localization in General Rings E. Bombieri and W. Gubler Heights in Diophantine Geometry Y. J. Ionin and M. S. Shrikhande Combinatorics of Symmetric Designs S. Berhanu, P. D. Cordaro and J. Hounie An Introduction to Involutive Structures A. Shlapentokh Hilbert’s Tenth Problem G. Michler Theory of Finite Simple Groups I A. Baker and G. W¨ustholz Logarithmic Forms and Diophantine Geometry P. Kronheimer and T. Mrowka Monopoles and Three-Manifolds B. Bekka, P. de la Harpe and A. Valette Kazhdan’s Property (T) J. Neisendorfer Algebraic Methods in Unstable Homotopy Theory M. Grandis Directed Algebraic Topology G. Michler Theory of Finite Simple Groups II R. Schertz Complex Multiplication S. Bloch Lectures on Algebraic Cycles (2nd Edition) B. Conrad, O. Gabber and G. Prasad Pseudo-reductive Groups T. Downarowicz Entropy in Dynamical Systems C. Simpson Homotopy Theory of Higher Categories E. Fricain and J. Mashreghi The Theory of H(b) Spaces I E. Fricain and J. Mashreghi The Theory of H(b) Spaces II J. Goubault-Larrecq Non-Hausdorff Topology and Domain Theory ´ J. Sniatycki Differential Geometry of Singular Spaces and Reduction of Symmetry E. Riehl Categorical Homotopy Theory B. A. Munson and I. Voli´c Cubical Homotopy Theory B. Conrad, O. Gabber and G. Prasad Pseudo-reductive Groups (2nd Edition) J. Heinonen, P. Koskela, N. Shanmugalingam and J. T. Tyson Sobolev Spaces on Metric Measure Spaces Y.-G. Oh Symplectic Topology and Floer Homology I Y.-G. Oh Symplectic Topology and Floer Homology II A. Bobrowski Convergence of One-Parameter Operator Semigroups K. Costello and O. Gwilliam Factorization Algebras in Quantum Field Theory I J.-H. Evertse and K. Gy˝ory Discriminant Equations in Diophantine Number Theory

Factorization Algebras in Quantum Field Theory Volume 1

K E V I N CO S T E L L O Perimeter Institute for Theoretical Physics, Waterloo, Ontario OWE N G W I L L I A M Max Planck Institute for Mathematics, Bonn

University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 4843/24, 2nd Floor, Ansari Road, Daryaganj, Delhi - 110002, India 79 Anson Road, #06-04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781107163102 10.1017/9781316678626 c Kevin Costello and Owen Gwilliam 2017  This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2017 A catalogue record for this publication is available from the British Library ISBN 978-1-107-16310-2 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party Internet Web sites referred to in this publication and does not guarantee that any content on such Web sites is, or will remain, accurate or appropriate.

T O L AUREN AND T O S OPHIE

Contents

1

Introduction 1.1 The Motivating Example of Quantum Mechanics 1.2 A Preliminary Definition of Prefactorization Algebras 1.3 Prefactorization Algebras in Quantum Field Theory 1.4 Comparisons with Other Formalizations of Quantum Field Theory 1.5 Overview of This Volume 1.6 Acknowledgments PART I

PREFACTORIZATION ALGEBRAS

page 1 3 8 8 11 16 18 21

2

From Gaussian Measures to Factorization Algebras 2.1 Gaussian Integrals in Finite Dimensions 2.2 Divergence in Infinite Dimensions 2.3 The Prefactorization Structure on Observables 2.4 From Quantum to Classical 2.5 Correlation Functions 2.6 Further Results on Free Field Theories 2.7 Interacting Theories

23 25 27 31 34 36 39 40

3

Prefactorization Algebras and Basic Examples 3.1 Prefactorization Algebras 3.2 Associative Algebras from Prefactorization Algebras on R 3.3 Modules as Defects 3.4 A Construction of the Universal Enveloping Algebra 3.5 Some Functional Analysis

44 44 51 52 59 62

vii

viii

Contents

3.6 3.7

4

5

6

7

The Factorization Envelope of a Sheaf of Lie Algebras Equivariant Prefactorization Algebras

73 79

PART II FIRST EXAMPLES OF FIELD THEORIES AND THEIR OBSERVABLES

87

Free Field Theories 4.1 The Divergence Complex of a Measure 4.2 The Prefactorization Algebra of a Free Field Theory 4.3 Quantum Mechanics and the Weyl Algebra 4.4 Pushforward and Canonical Quantization 4.5 Abelian Chern–Simons Theory 4.6 Another Take on Quantizing Classical Observables 4.7 Correlation Functions 4.8 Translation-Invariant Prefactorization Algebras 4.9 States and Vacua for Translation Invariant Theories

89 89 93 106 112 115 124 129 131 139

Holomorphic Field Theories and Vertex Algebras 5.1 Vertex Algebras and Holomorphic Prefactorization Algebras on C 5.2 Holomorphically Translation-Invariant Prefactorization Algebras 5.3 A General Method for Constructing Vertex Algebras 5.4 The βγ System and Vertex Algebras 5.5 Kac–Moody Algebras and Factorization Envelopes

145

157 171 188

PART III FACTORIZATION ALGEBRAS

205

Factorization Algebras: Definitions and Constructions 6.1 Factorization Algebras 6.2 Factorization Algebras in Quantum Field Theory 6.3 Variant Definitions of Factorization Algebras 6.4 Locally Constant Factorization Algebras 6.5 Factorization Algebras from Cosheaves 6.6 Factorization Algebras from Local Lie Algebras

207 207 215 216 220 225 230

Formal Aspects of Factorization Algebras 7.1 Pushing Forward Factorization Algebras 7.2 Extension from a Basis

232 232 232

145 149

Contents

7.3 7.4 8

Pulling Back Along an Open Immersion Descent Along a Torsor

Factorization Algebras: Examples 8.1 Some Examples of Computations 8.2 Abelian Chern–Simons Theory and Quantum Groups

ix

240 241 243 243 249

Appendix A Background

273

Appendix B Functional Analysis

310

Appendix C Homological Algebra in Differentiable Vector Spaces

351

Appendix D The Atiyah–Bott Lemma

374

References

377

Index

383

1 Introduction

This two-volume book provides the analog, in quantum field theory, of the deformation quantization approach to quantum mechanics. In this introduction, we start by recalling how deformation quantization works in quantum mechanics. The collection of observables in a quantum mechanical system forms an associative algebra. The observables of a classical mechanical system form a Poisson algebra. In the deformation quantization approach to quantum mechanics, one starts with a Poisson algebra Acl and attempts to construct an associative algebra Aq , which is an algebra flat over the ring C[[h]], ¯ together with an isomorphism of associative algebras Aq /h¯ ∼ = Acl . In addition, if a, b ∈ Acl , and  a,  b are any lifts of a, b to Aq , then 1  [ a, b] = {a, b} ∈ Acl . h¯ →0 h ¯ lim

Thus, Acl is recovered in the h¯ → 0 limit, i.e., the classical limit. We will describe an analogous approach to studying perturbative quantum field theory. To do this, we need to explain the following. • The structure present on the collection of observables of a classical field theory. This structure is the analog, in the world of field theory, of the commutative algebra that appears in classical mechanics. We call this structure a commutative factorization algebra. • The structure present on the collection of observables of a quantum field theory. This structure is that of a factorization algebra. We view our definition of factorization algebra as a differential geometric analog of a definition introduced by Beilinson and Drinfeld. However, the definition we use is very closely related to other definitions in the literature, in particular to the Segal axioms. 1

2

Introduction

• The additional structure on the commutative factorization algebra associated to a classical field theory that makes it “want” to quantize. This structure is the analog, in the world of field theory, of the Poisson bracket on the commutative algebra of observables. • The deformation quantization theorem we prove. This states that, provided certain obstruction groups vanish, the classical factorization algebra associated to a classical field theory admits a quantization. Further, the set of quantizations is parametrized, order by order in h¯ , by the space of deformations of the Lagrangian describing the classical theory.

This quantization theorem is proved using the physicists’ techniques of perturbative renormalization, as developed mathematically in Costello (2011b). We claim that this theorem is a mathematical encoding of the perturbative methods developed by physicists. This quantization theorem applies to many examples of physical interest, including pure Yang–Mills theory and σ -models. For pure Yang–Mills theory, it is shown in Costello (2011b) that the relevant obstruction groups vanish and that the deformation group is one-dimensional; thus there exists a one-parameter family of quantizations. In Li and Li (2016), the topological B-model with target a complex manifold X is constructed; the obstruction to quantization is that X be Calabi–Yau. Li and Li show that the observables and correlations functions recovered by their quantization agree with well-known formulas. Grady, Li, and Li (2015) describe a one-dimensional σ -model with a smooth symplectic manifold as target and show how it recovers Fedosov quantization. Other examples are considered in Costello (2010, 2011a), Costello and Li (2011), and Gwilliam and Grady (2014). We will explain how (under certain additional hypotheses) the factorization algebra associated to a perturbative quantum field theory encodes the correlation functions of the theory. This fact justifies the assertion that factorization algebras encode a large part of quantum field theory. This work is split into two volumes. Volume 1 develops the theory of factorization algebras and explains how the simplest quantum field theories – free theories – fit into this language. We also show in this volume how factorization algebras provide a convenient unifying language for many concepts in topological and quantum field theory. Volume 2, which is more technical, derives the link between the concept of perturbative quantum field theory as developed in Costello (2011b) and the theory of factorization algebras.

1.1 The Motivating Example of Quantum Mechanics

3

1.1 The Motivating Example of Quantum Mechanics The model problems of classical and quantum mechanics involve a particle moving in some Euclidean space Rn under the influence of some fixed field. Our main goal in this section is to describe these model problems in a way that makes the idea of a factorization algebra (Section 6.1 in Chapter 6) emerge naturally, but we also hope to give mathematicians some sense of the physical meaning of terms such as “field” and “observable.” We will not worry about making precise definitions, since that’s what this book aims to do. As a narrative strategy, we describe a kind of cartoon of a physical experiment, and we ask that physicists accept this cartoon as a friendly caricature elucidating the features of physics we most want to emphasize.

1.1.1 A Particle in a Box For the general framework we want to present, the details of the physical system under study are not so important. However, for concreteness, we focus attention on a very simple system: that of a single particle confined to some region of space. We confine our particle inside some box and occasionally take measurements of this system. The set of possible trajectories of the particle around the box constitutes all the imaginable behaviors of this particle; we might describe this space of behaviors mathematically as Maps(I, Box), where I ⊂ R denotes the time interval over which we conduct the experiment. We say the set of possible behaviors forms a space of fields on the timeline of the particle. The behavior of our theory is governed by an action functional, which is a function on Maps(I, Box). The simplest case typically studied is the massless free field theory, whose value on a trajectory f : I → Box is 

(f (t), f¨ (t)) dt.

S(f ) = t∈I

Here we use (−, −) to denote the usual inner product on Rn , where we view the box as a subspace of Rn , and f¨ to denote the second derivative of f in the time variable t. The aim of this section is to outline the structure one would expect the observables – that is, the possible measurements one can make of this system – should satisfy.

4

Introduction

1.1.2 Classical Mechanics Let us start by considering the simpler case where our particle is treated as a classical system. In that case, the trajectory of the particle is constrained to be in a solution to the Euler–Lagrange equations of our theory, which is a differential equation determined by the action functional. For example, if the action functional governing our theory is that of the massless free theory, then a map f : I → Box satisfies the Euler–Lagrange equation if it is a straight line. (Since we are just trying to provide a conceptual narrative here, we will assume that Box becomes all of Rn so that we do not need to worry about what happens at the boundary of the box.) We are interested in the observables for this classical field theory. Since the trajectory of our particle is constrained to be a solution to the Euler–Lagrange equation, the only measurements one can make are functions on the space of solutions to the Euler–Lagrange equation. If U ⊂ R is an open subset, we will let Fields(U) denote the space of fields on U, that is, the space of maps f : U → Box. We will let EL(U) ⊂ Fields(U) denote the subspace consisting of those maps f : U → Box that are solutions to the Euler–Lagrange equation. As U varies, EL(U) forms a sheaf of spaces on R. We will let Obscl (U) denote the commutative algebra of functions on EL(U) (the precise class of functions we consider discussed later). We will think of Obscl (U) as the collection of observables for our classical system that depend only upon the behavior of the particle during the time period U. As U varies, the algebras Obscl (U) vary and together constitute a cosheaf of commutative algebras on R.

1.1.3 Measurements in Quantum Mechanics The notion of measurement is fraught in quantum theory, but we will take a very concrete view. Taking a measurement means that we have a physical measurement device (e.g., a camera) that we allow to interact with our system for a period of time. The measurement is then how our measurement device has changed due to the interaction. In other words, we couple the two physical systems so that they interact, then decouple them and record how the measurement device has modified from its initial condition. (Of course, there is a symmetry in this situation: both systems are affected by their interaction, so a measurement inherently disturbs the system under study.)

1.1 The Motivating Example of Quantum Mechanics

5

The observables for a physical system are all the imaginable measurements we could take of the system. Instead of considering all possible observables, we might also consider those observables that occur within a specified time period. This period can be specified by an open interval U ⊂ R. Thus, we arrive at the following principle. Principle 1. For every open subset U ⊂ R, we have a set Obs(U) of observables one can make during U. Our second principle is a minimal version of the linearity implied by, e.g., the superposition principle. Principle 2. The set Obs(U) is a complex vector space. We think of Obs(U) as being the collection of ways of coupling a measurement device to our system during the time period U. Thus, there is a natural map Obs(U) → Obs(V) if U ⊂ V is a shorter time interval. This means that the space Obs(U) forms a precosheaf.

1.1.4 Combining Observables Measurements (and so observables) differ qualitatively in the classical and quantum settings. If we study a classical particle, the system is not noticeably disturbed by measurements, and so we can do multiple measurements at the same time. (To be a little less sloppy, we suppose that by refining our measuring devices, we can make the impact on the particle as small as we would like.) Hence, on each interval J we have a commutative multiplication map Obs(J) ⊗ Obs(J) → Obs(J). We also have maps Obs(I) ⊗ Obs(J) → Obs(K) for every pair of disjoint intervals I, J contained in an interval K, as well as the maps that let us combine observables on disjoint intervals. For a quantum particle, however, a measurement typically disturbs the system significantly. Taking two measurements simultaneously is incoherent, as the measurement devices are coupled to each other and thus also affect each other, so that we are no longer measuring just the particle. Quantum observables thus do not form a cosheaf of commutative algebras on the interval. However, there are no such problems with combining measurements occurring at different times. Thus, we find the following. Principle 3. If U, U  are disjoint open subsets of R, and U, U  ⊂ V where V is also open, then there is a map  : Obs(U) ⊗ Obs(U  ) → Obs(V).

6

Introduction

If O ∈ Obs(U) and O ∈ Obs(U  ), then O  O is defined by coupling our system to measuring device O during the period U and to device O during the period U  . Further, there are maps for a finite collection of disjoint time intervals contained in a long time interval, and these maps are compatible under composition of such maps. (The precise meaning of these terms is detailed in Section 3.1 in Chapter 3.)

1.1.5 Perturbative Theory and the Correspondence Principle In the bulk of this two-volume book, we will be considering perturbative quantum theory. For us, this adjective “perturbative” means that we work over the base ring C[[h]], ¯ where at h¯ = 0 we find the classical theory. In perturbative theory, therefore, the space Obs(U) of observables on an open subset U is a C[[h]]-module, and the product maps are C[[h]]-linear. ¯ ¯ The correspondence principle states that the quantum theory, in the h¯ → 0 limit, must reproduce the classical theory. Applied to observables, this leads to the following principle. Principle 4. The vector space Obsq (U) of quantum observables is a flat C[[h]]¯ module such that modulo h¯ , it is equal to the space Obscl (U) of classical observables. These four principles are at the heart of our approach to quantum field theory. They say, roughly, that the observables of a quantum field theory form a factorization algebra, which is a quantization of the factorization algebra associated to a classical field theory. The main theorem presented in this twovolume book is that one can use the techniques of perturbative renormalization to construct factorization algebras perturbatively quantizing a certain class of classical field theories (including many classical field theories of physical and mathematical interest). As we have mentioned, this first volume focuses on the general theory of factorization algebras and on simple examples of field theories; this result is derived in Volume 2.

1.1.6 Associative Algebras in Quantum Mechanics The principles we have described so far indicate that the observables of a quantum mechanical system should assign, to every open subset U ⊂ R, a vector space Obs(U), together with a product map Obs(U) ⊗ Obs(U  ) → Obs(V)

1.1 The Motivating Example of Quantum Mechanics

7

if U, U  are disjoint open subsets of an open subset V. This is the basic data of a factorization algebra (see Section 3.1 in Chapter 3). It turns out that in the case of quantum mechanics, the factorization algebra produced by our quantization procedure has a special property: it is locally constant (see Section 6.4 in Chapter 6). This means that the map Obs((a, b)) → Obs(R) is an isomorphism for any interval (a, b). Let A denote the vector space Obs(R); note that A is canonically isomorphic to Obs((a, b)) for any interval (a, b). The product map Obs((a, b)) ⊗ Obs((c, d)) → Obs((a, d)) when a < b < c < d, becomes, via this isomorphism, a product map m : A ⊗ A → A. The axioms of a factorization algebra imply that this multiplication turns A into an associative algebra. As we will see in Section 4.2 in Chapter 4, this associative algebra is the Weyl algebra, which one expects to find as the algebra of observables for quantum mechanics of a particle moving in Rn . This kind of geometric interpretation of algebra should be familiar to topologists: associative algebras are algebras over the operad of little intervals in R, and this is precisely what we have described. As we explain in Section 6.4 in Chapter 6, this relationship continues and so our quantization theorem produces many new examples of algebras over the operad En of little n-discs. An important point to take away from this discussion is that associative algebras appear in quantum mechanics because associative algebras are connected with the geometry of R. There is no fundamental connection between associative algebras and any concept of “quantization”: associative algebras appear only when one considers one-dimensional quantum field theories. As we will see later, when one considers topological quantum field theories on n-dimensional space–times, one finds a structure reminiscent of an En -algebra instead of an E1 -algebra. Remark: As a caveat to the strong assertion in the preceding (and jumping ahead of our story), note that for a manifold of the form X → R, one can push forward a factorization algebra Obs on X × R to a factorization algebra π∗ Obs on R along the projection map π : X × R → R. In this case, π∗ Obs((a, b)) = Obs(X × (a, b)). Hence, a quantization of a higher dimensional theory will produce, via such pushforwards to R, deformations of associative algebras, but knowing only the pushforward is typically insufficient to reconstruct the factorization algebra on the higher dimensional manifold. ♦

8

Introduction

1.2 A Preliminary Definition of Prefactorization Algebras Below (see Section 3.1 in Chapter 3) we give a more formal definition, but here we provide the basic idea. Let M be a topological space (which, in practice, will be a smooth manifold). Definition 1.2.1 A prefactorization algebra F on M, taking values in cochain complexes, is a rule that assigns a cochain complex F(U) to each open set U ⊂ M along with (i) A cochain map F(U) → F(V) for each inclusion U ⊂ V. (ii) A cochain map F(U1 )⊗· · ·⊗F(Un ) → F(V) for every finite collection of open sets where each Ui ⊂ V and the Ui are disjoint. (iii) The maps are compatible in a certain natural way. The simplest case of this compatibility is that if U ⊂ V ⊂ W is a sequence of open sets, the map F(U) → F(W) agrees with the composition through F(V)). Remark: A prefactorization algebra resembles a precosheaf, except that we tensor the cochain complexes rather than taking their direct sum. ♦ The observables of a field theory, whether classical or quantum, form a prefactorization algebra on the space–time manifold M. In fact, they satisfy a kind of local-to-global principle in the sense that the observables on a large open set are determined by the observables on small open sets. The notion of a factorization algebra (Section 6.1 in Chapter 6) makes this local-to-global condition precise.

1.3 Prefactorization Algebras in Quantum Field Theory The (pre)factorization algebras of interest in this book arise from perturbative quantum field theories. We have already discussed in Section 1.1 how factorization algebras appear in quantum mechanics. In this section we will see how this picture extends in a natural way to quantum field theory. The manifold M on which the prefactorization algebra is defined is the space–time manifold of the quantum field theory. If U ⊂ M is an open subset, we will interpret F(U) as the collection of observables (or measurements) that we can make that depend only on the behavior of the fields on U. Performing a measurement involves coupling a measuring device to the quantum system in the region U. One can bear in mind the example of a particle accelerator. In that situation, one can imagine the space–time M as being of the form M = A × (0, t), where A is the interior of the accelerator and t is the duration of our experiment.

1.3 Prefactorization Algebras in Quantum Field Theory

9

In this situation, performing a measurement on an open subset U ⊂ M is something concrete. Let us take U = V × (ε, δ), where V ⊂ A is some small region in the accelerator and where (ε, δ) is a short time interval. Performing a measurement on U amounts to coupling a measuring device to our accelerator in the region V, starting at time ε and ending at time δ. For example, we could imagine that there is some piece of equipment in the region V of the accelerator, which is switched on at time ε and switched off at time δ.

1.3.1 Interpretation of the Prefactorization Algebra Axioms Suppose that we have two different measuring devices, O1 and O2 . We would like to set up our accelerator so that we measure both O1 and O2 . There are two ways we can do this. We can insert O1 and O2 into disjoint regions V1 , V2 of our accelerator. Then we can turn O1 and O2 on at any times we like, including for overlapping time intervals. If the regions V1 , V2 overlap, then we cannot do this. After all, it doesn’t make sense to have two different measuring devices at the same point in space at the same time. However, we could imagine inserting O1 into region V1 during the time interval (a, b); and then removing O1 , and inserting O2 into the overlapping region V2 for the disjoint time interval (c, d). These simple considerations immediately suggest that the possible measurements we can make of our physical system form a prefactorization algebra. Let Obs(U) denote the space of measurements we can make on an open subset U ⊂ M. Then, by combining measurements in the way outlined in the preceding text, we would expect to have maps Obs(U) ⊗ Obs(U  ) → Obs(V) whenever U, U  are disjoint open subsets of an open subset V. The associativity and commutativity properties of a prefactorization algebra are evident.

1.3.2 The Cochain Complex of Observables In the approach to quantum field theory considered in this book, the factorization algebra of observables will be a factorization algebra of cochain complexes. That is, Obs assigns a cochain complex Obs(U) to each open U. One can ask for the physical meaning of the cochain complex. We will repeatedly mention observables in a gauge theory, as these kinds of cohomological aspects are well known for such theores.

10

Introduction

It turns out that the “physical” observables will be H 0 (Obs(U)). If O ∈ Obs0 (U) is an observable of cohomological degree 0, then the equation O. = 0 can often be interpreted as saying that O is compatible with the gauge symmetries of the theory. Thus, only those observables O ∈ Obs0 (U) that are closed are physically meaningful. The equivalence relation identifying O ∈ Obs0 (U) with O + O. , where O ∈ Obs−1 (U), also has a physical interpretation, which will take a little more work to describe. Often, two observables on U are physically indistinguishable (that is, they cannot be distinguished by any measurement one can perform). In the example of an accelerator outlined earlier, two measuring devices are equivalent if they always produce the same expectation values, no matter how we prepare our system, or no matter what boundary conditions we impose. As another example, in the quantum mechanics of a free particle, the observable measuring the momentum of a particle at time t is equivalent to that measuring the momentum of a particle at another time t . This is because, even at the quantum level, momentum is preserved (as the momentum operator commutes with the Hamiltonian). From the cohomological point of view, if O, O ∈ Obs0 (U) are both in the kernel of the differential (and thus “physically meaningful”), then they are equivalent in the sense described previously if they differ by an exact observable. It is a little more difficult to provide a physical interpretation for the other cohomology groups H n (Obs(U)). The first cohomology group H 1 (Obs(U)) contains anomalies (or obstructions) to lifting classical observables to the quantum level. For example, in a gauge theory, one might have a classical observable that respects gauge symmetry. However, it may not lift to a quantum observable respecting gauge symmetry; this happens if there is a nontrivial anomaly in H 1 (Obs(U)). The cohomology groups H n (Obs(U)), when n < 0, are best interpreted as symmetries, and higher symmetries, of observables. Indeed, we have seen that the physically meaningful observables are the closed degree 0 elements of Obs(U). One can construct a simplicial set, whose n-simplices are closed and degree 0 elements of Obs(U) ⊗ ∗ ( n ). The vertices of this simplicial set are observables, the edges are equivalences between observables, the faces are equivalences between equivalences, and so on. The Dold–Kan correspondence (see Theorem A.2.7 in Appendix A) tells us that the nth homotopy group of this simplicial set is H −n (Obs(U)). This allows us to interpret H −1 (Obs(U)) as being the group of symmetries of the trivial observable 0 ∈ H 0 (Obs(U)), and H −2 (Obs(U)) as the symmetries of the identity symmetry of 0 ∈ H 0 (Obs(U)), and so on.

1.4 Comparisons with Other Formalizations of Quantum Field Theory 11 Although the cohomology groups H n (Obs(U)) where n ≥ 1 do not have a clear physical interpretation as clear as that for H 0 , they are mathematically very natural objects and it is important not to discount them. For example, let us consider a gauge theory on a manifold M, and let D be a disc in M. Then it is often the case that elements of H 1 (Obs(D)) can be integrated over a circle in M to yield cohomological degree 0 observables (such as Wilson operators).

1.4 Comparisons with Other Formalizations of Quantum Field Theory Now that we have explained carefully what we mean by a prefactorization algebra, let us say a little about the history of this concept, and how it compares to other mathematical approaches to quantum field theory. We make no attempt to state formal theorems relating our approach to other axiom systems. Instead we sketch heuristic relationships between the various axiom systems.

1.4.1 Factorization Algebras in the Sense of Beilinson and Drinfeld For us, one source of inspiration is the work of Beilinson and Drinfeld on chiral conformal field theory. These authors gave a geometric reformulation of the theory of vertex algebras in terms of an algebro-geometric version of the concept of factorization algebra. For Beilinson and Drinfeld, a factorization algebra on an algebraic curve X is, in particular, a collection of sheaves Fn on the Cartesian powers X n of X. If (x1 , . . . , xn ) ∈ X n is an n-tuple of distinct points in X, let Fx1 ,...,xn denote the stalk of Fn at this point in X n . The axioms of Beilinson and Drinfeld imply that there is a canonical isomorphism Fx1 ,...,xn ∼ = F x 1 ⊗ · · · ⊗ Fx n . In fact, their axioms tell us that the restriction of the sheaf Fn to any stratum of X n (in the stratification by number of points) is determined by the sheaf F1 on X. The fundamental object in their approach is the sheaf F1 . All the other sheaves Fn are built from copies of F1 by certain gluing data, which we can think of as structures put on the sheaf F1 . One should think of the stalk Fx of F1 at x as the space of local operators in a field theory at the point x. Thus, F1 is the sheaf on X whose stalks are the spaces of local operators. The other structures on F1 reflect the operator product expansions of local operators.

12

Introduction

Let us now sketch, heuristically, how we expect their approach to be related  on X in our sense. Then, to ours. Suppose we have a factorization algebra F  for every open V ⊂ X, we have a vector space F(V) of observables on V. The space of local operators associated to a point x ∈ X should be thought of as those observables that live on every open neighbourhood of x. In other words, we can define  x = lim F(V) F x∈V

to be the limit over open neighbourhoods of x of the observables on that neigh bourhood. This limit is the costalk of the pre-cosheaf F. Thus, the heuristic translation between their axioms and ours is that the sheaf F1 on X that they construct should have stalks coinciding with the costalks of our factorization algebra. We do not know how to turn this idea into a precise theorem in general. We do, however, have a precise theorem in one special case. Beilinson and Drinfeld show that a factorization algebra in their sense on the affine line A1 , which is also translation and rotation equivariant, is the same as a vertex algebra. We have a similar theorem. We show in Chapter 5 that a factorization algebra on C that is translation and rotation invariant, and also has a certain “holomorphic” property, gives rise to a vertex algebra. Therefore, in this special case, we can show how a factorization algebra in our sense gives rise to one in the sense used by Beilinson and Drinfeld.

1.4.2 Segal’s Axioms for Quantum Field Theory Segal has developed and studied some very natural axioms for quantum field theory (see, e.g., Segal 2010). These axioms were first studied in the world of topological field theory by Atiyah, Segal, and Witten, and in conformal field theory by Kontsevich and Segal (2004). According to Segal’s philosophy, a d-dimensional quantum field theory (in of dEuclidean signature) is a symmetric functor from the category CobRiem d is a dimensional Riemannian cobordisms. An object of the category CobRiem d compact d − 1-manifold together with a germ of a d-dimensional Riemannian structure. A morphism is a d-dimensional Riemannian cobordism. The symmetric monoidal structure arises from disjoint union. As defined, this category does not have identity morphisms, but they can be added in formally. Definition 1.4.1 A Segal field theory is a symmetric monoidal functor from to the category of (topological) vector spaces. CobRiem d

1.4 Comparisons with Other Formalizations of Quantum Field Theory 13

We won’t get into details about what kind of topological vector spaces one should consider, because our aim is just to sketch a heuristic relationship between Segal’s picture and our picture. In our approach to studying quantum field theory, the fundamental objects are not the Hilbert spaces associated to codimension 1 manifolds, but rather the spaces of observables. Any reasonable axiom system for quantum field theory should be able to capture the notion of observable. In particular, we should be able to understand observables in terms of Segal’s axioms. Segal, in lectures and conversations, has explained how to do this. Suppose we have a Riemannian manifold M and a point x ∈ M. Consider a ball B(x, r) of radius r around x, whose boundary is a sphere S(x, r). Segal explains that the Hilbert space Z(S(x, r)) should be thought of as the space of operators on the ball B(x, r). If r < r , there is a cobordism S(x, r) → S(x, r ) given by the complement of B(x, r) in the closed ball B(x, r ). This gives rise to maps Z(S(x, r)) → Z(S(x, r )). Segal defines the space of local operators at x to be the limit lim Z(S(x, r))

r→0

of this inverse system. One can understand from this idea of Segal’s how one should construct something like a prefactorization algebra on any Riemannian manifold M of dimension n from a Segal field theory. Given an open subset U ⊂ M whose boundary is a codimension 1 submanifold ∂U, we define the space F(U) of observables on U to be Z(∂U). If U, V, W are three such opens in M, such that the closures of U and V are disjoint in W, then there is a cobordism W \ (U  V) : ∂U  ∂V → ∂W. This cobordism induces a map F(U) ⊗ F(V) = Z(∂U) ⊗ Z(∂V) → Z(∂W) = F(W). There are similar maps defined when U1 , . . . , Un , W are opens with smooth boundary such that the closures of the Ui are disjoint and contained in W. In this way, we can construct from a Segal field theory something that is very like a prefactorization algebra; the only difference is that we restrict our attention to those opens with smooth boundary, and the prefactorization algebra structure maps are defined only for collections of opens whose closures are disjiont. Remark: In the text that follows we discuss how certain universal factorization algebras relate to topological field theories in the style of Atiyah–Segal–Lurie. Dwyer, Stolz, and Teichner have also proposed an approach to constructing

14

Introduction

Segal-style non-topological field theories, such as Riemannian field theories, using factorization algebras. ♦

1.4.3 Topological Field Theory One class of field theories for which there exists an extensive mathematical literature is topological field theories (see, e.g., Lurie 2009c). One can ask how our axiom system relates to those for topological field theories. There is a subclass of factorization algebras that appear in topological field theories, called locally constant factorization algebras. A factorization algebra F on a manifold M, valued in cochain complexes, is locally constant if, for any two discs D1 ⊂ D2 in M, the map F(D1 ) → F(D2 ) is a quasi-isomorphism. A theorem of Lurie (2016) shows that, given a locally constant factorization algebra F on Rn , the complex F(D) has the structure of an En algebra. This relationship matches with what one expects from the standard approach to the axiomatics of topological field theory. According to the standard axioms for topological field theories, a topological field theory (TFT) of dimension n is a symmetric monoidal functor Z : Cobn → Vect from the n-dimensional cobordism category to the category of vector spaces. Here, Cobn is the category whose objects are closed smooth n − 1-manifolds and the morphisms are cobordisms between them. (It is standard, following Freed (1994), to consider also higher-categorical objects associated to manifolds of higher codimension.) For an n−1-manifold N, we should interpret Z(N) as the Hilbert space of the TFT on N. Then, following standard physics yoga, we should interpret Z(Sn−1 ) as the space of local operators of the theory. There are natural cobordisms between disjoint unions of the n − 1-sphere that make Z(Sn−1 ) into an En algebra. For example, in dimension 2, the pair of pants with k legs provides the k-ary operations for the E2 algebra structure on Z(S1 ). This story fits nicely with our approach. If we have a locally constant factorization algebra F on Rn , then F(Dn ) is an En algebra. Further, we interpret F(Dn ) as being the space of observables supported on an n-disc. Since F is locally constant, this may as well be the observables supported on a point, because it is independent of the radius of the n-disc.

1.4.4 Correlation Functions In some classic approaches to the axiomatics of quantum field theory – such as the Wightman axioms or the Osterwalder–Schrader axioms, their Euclidean

1.4 Comparisons with Other Formalizations of Quantum Field Theory 15

counterpart – the fundamental objects are correlation functions. While we make no attempt to verify that a factorization algebra gives rise to a solution of any of these axiom systems, we do show that the factorization algebra has enough data to define the correlation functions. Let us briefly explain how this works for two different classes of examples. Suppose that we have a factorization algebra F on a manifold M over the ring R[[h]] ¯ of formal power series in h. ¯ Suppose that H 0 (F(M)) = R[[h]]. ¯ This condition holds in some natural examples: for instance, for Chern–Simons theory on R3 or for a massive scalar field theory on a compact manifold M. Let U1 , . . . , Un ⊂ M be disjoint open sets. The factorization product gives us a R[[h]]-multilinear map ¯ − : H 0 (F(U1 )) × · · · × H 0 (F(Un )) → H 0 (F(M)) = R[[h]]. ¯ In this way, given observables Oi ∈ H 0 (F(Ui )), we can produce a formal power series in h¯ O1 , . . . , On ∈ R[[h]]. ¯ In the field theories we just mentioned, this map does compute the expectation value of observables. (See Section 4.6 in Chapter 4, where we describe this map in terms of the Green’s functions. We also recover the Gauss linking number there from Abelian Chern–Simons theory.) This construction doesn’t give us expectation values in every situation where we might hope to construct them. For example, if we work with a field theory on Rn , it is rarely the case that H 0 (F(Rn )) is isomorphic to R[[h]]. ¯ We would, however, expect to be able to define correlation functions in this situation. To achieve this, we define a variant of this construction that works well on Rn . Given a factorization algebra on Rn with ground ring R[[h]] ¯ as before, we define in Section 4.9 in Chapter 4 a vacuum to be an R[[h]]-linear map ¯ − : H 0 (F(Rn )) → R[[h]] ¯ that is translation-invariant and satisfies a certain cluster decomposition principle. After choosing a vacuum, one can define correlation functions in the same way as in the preceding text.

16

Introduction

1.5 Overview of This Volume This two-volume work concerns, as the titles suggests, factorization algebras and quantum field theories. In this introduction so far, we have sketched what a prefactorization algebra is and why it might help organize and analyze the behavior of the observables of a quantum field theory. These two volumes develop these ideas further in a number of ways. The first volume focuses on factorization algebras: their definition, some formal properties, and some simple constructions of factorization algebras. The quantum field theory in this volume is mostly limited to free field theories. In a moment, we give a detailed overview of this volume. In the second volume, we focus on our core project: we develop the Batalin– Vilkovisky formalism for both classical and quantum field theories and show how it automatically produces a deformation quantization of factorization algebras of observables. In particular, Volume 2 will introduce the factorization algebras associated to interacting field theories. We also provide there a refinement of the Noether theorems in the setting of factorization algebras, in which, roughly speaking, local symmetries of a field theory lift to a map of factorization algebras,. This map realizes the symmetries as observables of the field theory. For a more detailed overview of Volume 2, see its introductory chapter.

1.5.1 Chapter by Chapter Chapter 2 serves as a second introduction. In this chapter we explain, using informal language and without any background knowledge required, how the observables of a free scalar field theory naturally form a prefactorization algebra. Readers who want to understand the main ideas of this two-volume work with the minimum amount of technicalities should start there. Chapter 3 gives a more careful definition of the concept of a prefactorization algebra, and analyzes some basic examples. In particular, the relationship between prefactorization algebras on R and associative algebras is developed in detail. We also introduce a construction that will play an important role in the rest of the book: the factorization envelope of a sheaf of Lie algebras on a manifold. This construction is the factorization version of the universal enveloping algebra of a Lie algebra. In Chapter 4 we revisit the prefactorization algebras associated to a free field theory, but with more care and in greater generality than we used in Chapter 2. The methods developed in this chapter apply to gauge theories, treating with gauge symmetry with the method of Becchi, Rouet, Stora, and

1.5 Overview of This Volume

17

Tyutin (BRST) and its extension by Batalin and Vilkovisky. We analyze in some detail the example of Abelian Chern–Simons theory, and verify that the expectation value of Wilson lines in this theory recovers the Gauss linking number. Chapter 5 introduces the concept of a holomorphic prefactorization algebra on Cn . The prefactorization algebra of observables of a field theory with a holomorphic origin – such as a holomorphic twist Costello (2013a) of a supersymmetric gauge theory – will be such a holomorphic factorization algebra. We prove that a holomorphic prefactorization algebra on C gives rise to a vertex algebra, thus linking our story with a more traditional point of view on the algebra of operators of a chiral conformal field theory. The final chapters in this book develop the concept of factorization algebra, by adding a certain local-to-global axiom to the definition of prefactorization algebra. In Chapter 6, we provide the definition, discuss the relation between locally constant factorization algebras and En algebras, and explain how to construct several large classes of examples. In Chapter 7, we develop some formal properties of the theory of factorization algebras. Finally, in Chapter 8, we move beyond the formal and analyze some interesting explicit examples. For instance, we compute the factorization homology of the Kac–Moody enveloping factorization algebras, and we explain how Abelian Chern–Simons theory produces a quantum group.

1.5.2 A Comment on Functional Analysis and Algebra This book uses an unusual array of mathematical techniques, including both homological algebra and functional analysis. The homological algebra appears because our factorization algebras live in the world of cochain complexes (ultimately, because they come from the BV formalism for field theory). The functional analysis appears because our factorization algebras are built from vector spaces of an analytic nature, such as the space of distributions on a manifold. We have included an expository introduction to the techniques we use from homological algebra, operads, and sheaf theory in Appendix A. It is well known that it is hard to make homological algebra and functional analysis work well together. One reason is that, traditionally, the vector spaces that arise in analysis are viewed as topological vector spaces, and the category of topological vector spaces is not an Abelian category. In Appendix B, we introduce the concept of differentiable vector spaces. Differentiable vector spaces are more flexible than topological vector spaces, yet retain enough analytic structure for our purposes. We show that the category of differentiable vector spaces is an Abelian category, and indeed satisfies the strongest version

18

Introduction

of the axioms of an Abelian category: it is a locally presentable AB5 category. This means that homological algebra in the category of differentiable vector spaces works very nicely. We develop this in Appendix C. A gentle introduction to differentiable vector spaces, containing more than enough to follow everything in both volumes, is contained in Chapter 3, Section 3.5.

1.6 Acknowledgments The project of writing this book has stretched over many more years than we ever anticipated. In that course of time, we have benefited from conversations with many people, the chance to present this work at several workshops and conferences, and the feedback of various readers. The book and our understanding of this material is much better due to the interest and engagement of so many others. Thank you. We would like to thank directly the following people, although this list is undoubtedly incomplete: David Ayala, David Ben-Zvi, Dan Berwick-Evans, Damien Calaque, Alberto Cattaneo, Lee Cohn, Ivan Contreras, Vivek Dhand, Chris Douglas, Chris Elliott, John Francis, Dennis Gaitsgory, Sachin Gautam, Ezra Getzler, Greg Ginot, Ryan Grady, Andr´e Henriques, Rune Haugseng, Theo Johnson-Freyd, David Kazhdan, Si Li, Jacob Lurie, Takuo Matsuoka, Aaron Mazel-Gee, Pavel Mnev, David Nadler, Thomas Nikolaus, Fred Paugam, Dmitri Pavlov, Toly Preygel, Kasia Rejzner, Kolya Reshetikhin, Nick Rozenblyum, Claudia Scheimbauer, Graeme Segal, Thel Seraphim, Yuan Shen, Jim Stasheff, Stephan Stolz, Dennis Sullivan, Matt Szczesny, Hiro Tanaka, Peter Teichner, David Treumann, Philsang Yoo, Brian Williams, and Eric Zaslow for helpful conversations. K. C. is particularly grateful to Graeme Segal for many illuminating conversations about quantum field theory over the years. O.G. would like to thank the participants in the Spring 2014 Berkeley seminar and Fall 2014 seminar at the Max Planck Institute for Mathematics for their interest and extensive feedback; he is also grateful to Rune Haugseng and Dmitri Pavlov for help at the intersection of functional analysis and category theory. Peter Teichner’s feedback after teaching a Berkeley course in Spring 2016 on this material improved the book considerably. Finally, we are both grateful to John Francis and Jacob Lurie for introducing us to factorization algebras, in their topological incarnation, in 2008. Our work has taken place at many institutions – Northwestern University, the University of California, Berkeley, the Max Planck Institute for Mathematics, and the Perimeter Institute for Theoretical Physics – which provided

1.6 Acknowledgments

19

a supportive environment for writing and research. Research at the Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Economic Development and Innovation, and K. C.’s research at the Perimeter Institute is supported by the Krembil Foundation. We have also benefitted from the support of the National Science Foundation (NSF): K. C. and O. G. were partially supported by NSF grants DMS 0706945 and DMS 1007168, and O. G. was supported by NSF postdoctoral fellowship DMS-1204826. K. C. was also partially supported by a Sloan fellowship. Finally, during the period of this project, we have depended on the unrelenting support and love of our families. In fact, we have both gotten married and had children over these years! Thank you Josie, Dara, and Laszlo for being models of insatiable curiosity, ready sources of exuberant distraction, and endless fonts of joy. Thank you Lauren and Sophie—amidst all the tumult and the moves—for always encouraging us, for always sharing your wit and wisdom and warmth. It makes all the difference.

PA RT I Prefactorization algebras

2 From Gaussian Measures to Factorization Algebras

This chapter serves as a kind of second introduction, demonstrating how the free scalar field theory on a manifold M produces a factorization algebra on M. We have already sketched in Chapter 1 why the observables of a field theory ought to form a factorization algebra, without making precise what we meant by a quantum field theory. Here we will meet the second main theme of the book – the Batalin–Vilkovisky (BV) formalism for field theory – and see how it naturally produces a factorization algebra. Our approach to quantum field theory grows out of the idea of a path integral. Instead of trying to define such an integral directly, however, the BV formalism provides a homological approach to integration, similar in spirit to the de Rham complex. (As we will see in the next section, in the finite dimensional setting, the BV complex is isomorphic to the de Rham complex.) The philosophy goes like this. If the desired path integral were well defined mathematically, we could compute the expectation values of observables, and the expectation value map E is linear, so we obtain a linear equivalence relation between observables O ∼ O whenever E(O − O ) = 0. We can reconstruct, in fact, the expectation value by describing the inclusion RelE := ker E → Obs and taking the cokernel of this inclusion. In other words, we identify “integrands with the same integral.” The BV formalism approaches the problem from the other direction: even though the desired path integral may not be well defined, we often know, from physical arguments, when two observables ought to have the same expectation value (e.g., via Ward identities), so that we can produce a subspace RelBV → Obs. The BV formalism produces a subspace RelBV determined by the classical field theory, as we will see later, but this subspace is typically not of codimension 1. Further input, such as boundary conditions, is often necessary to get a number (i.e., to produce a codimension 1 subspace of relations). Nonetheless, the relations in RelBV would hold for

23

24

From Gaussian Measures to Factorization Algebras

any such choice, so any expectation value map coming from physics would factor through Obs /RelBV . In fact, the BV formalism produces a cochain complex, encoding relations between the relations and so on, whose zeroth cohomology group is the space Obs /RelBV . (Here, we are using Obs to denote the “naive” observables that one would first write down for the theory, not observables involving the auxiliary “anti-fields” introduced when applying the BV formalism.) That description is quite abstract; the rest of this chapter is about making the idea concrete with examples. In physics, a free field theory is one in which the action functional is a purely quadratic function of the fields. A basic example is the free scalar field theory on a Riemannian manifold (M, g), where the space of fields is the space C∞ (M) of smooth functions on M, and the action funtional is  1 φ(g + m2 )φ dvolg . S(φ) = 2 M

Here g refers to the Laplacian with the convention that its eigenvalues are nonnegative, and dvolg denotes the Riemannian volume form associated to the metric. The positive real number m is the mass of the theory. The main quantities of interest in the free field theory are the correlation functions, defined by the heuristic expression  φ(x1 ) · · · φ(xn ) = φ(x1 ) . . . φ(xn ) e−S(φ) dφ, φ∈C∞ (M)

where x1 , . . . , xn are points in M. Note that there is an observable O(x1 , . . . , xn ) : φ → φ(x1 ) · · · φ(xn ) sending a field φ to the product of its values at those points. The support of this observable is precisely the set of points {x1 , . . . , xn }, so this observable lives in Obs(U) for any open U containing all those points. The standard computations in quantum field theory tell us that this observable has the same expectation value as linear combinations of other correlation functions. For instance, Wick’s lemma tells us how the two-point correlation function relates to the Green’s function for g + m2 . Our task is to explain how the combination of the BV formalism and prefactorization algebras provides a simple and natural way to make sense of these relations. We will see that for a free field theory on a manifold M, there is a space of observables associated to any open subset U ⊂ M. We will see that the operations we can perform on these spaces of observables give us the structure of a prefactorization algebra on M. This example will serve as further

2.1 Gaussian Integrals in Finite Dimensions

25

motivation for the idea that observables of a field theory are described by a prefactorization algebra.

2.1 Gaussian Integrals in Finite Dimensions As in many approaches to quantum field theory, we will motivate our definition of the prefactorization algebra of observables by studying finite dimensional Gaussian integrals. Thus, let A be an n × n symmetric positive-definite real matrix, and consider Gaussian integrals of the form     xi Aij xj f (x) dn x exp − 12 x∈Rn

where f is a polynomial function on Rn . Note the formal analogy to the corre lation function we wish to compute: here x replaces φ, 12 xi Aij xj is quadratic in x as S is in φ, and f (x) is polynomial in x as O(φ) is a polynomial in φ. At this point, most textbooks on quantum field theory would explain Wick’s lemma, which is a combinatorial expression for such integrals. It reduces the integral above to a sum involving the quadratic moments of the Gaussian measure. Then, such a textbook would go on to define similar infinitedimensional Gaussian integrals using the analogous combinatorial expression. The key point is the simplicity of the moments of a Gaussian measure, which allows immediate generalization to infinite dimensions. We will take a different approach, however. Instead of focusing on the combinatorial expression for the integral, we will focus on the divergence operator associated to the Gaussian measure. (This operator provides the inclusion map Rel → Obs discussed at the beginning of this chapter.) Let P(Rn ) denote the space of polynomial functions on Rn . Let Vect(Rn ) denote the space of vector fields on Rn with polynomial coefficients. If dn x denotes the Lebesgue measure on Rn , let ωA denote the measure    ωA = exp − 12 xi Aij xj dn x. Then, the divergence operator DivωA associated to this measure is a linear map DivωA : Vect(Rn ) → P(Rn ), defined abstractly by saying that if V ∈ Vect(Rn ), then  LV ωA = DivωA V ωA

26

From Gaussian Measures to Factorization Algebras

where LV refers to the Lie derivative. Thus, the divergence of V measures the infinitesimal change in volume that arises when one applies the infinitesimal diffeomorphism V. In coordinates, the divergence is given by the formula

   ∂fi  ∂ fi fi xj Aij + . (†) =− DivωA ∂xi ∂xi i,j

i

(This formula is an exercise in applying Cartan’s magic formula: LV = [,.ιV ]. Note that this divergence operator is therefore a disguised version of the exterior derivative.) By the definition of divergence, we see   DivωA V ωA = 0 for all polynomial vector fields V, because     DivωA V ωA = LV ωA = (.ιV ωA ) and then we apply Stokes’ lemma. By changing basis of Rn to diagonalize A, one sees that the image of the divergence map is a codimension 1 linear subspace of the space P(Rn ) of polynomials on Rn . (This statement is true as long as A is nondegenerate; positive-definiteness is not required). Let us identify P(Rn )/ Im DivωA with R by taking the basis of the quotient space to be the image of the polynomial function 1. What we have shown so far can be summarized in the following lemma. Lemma 2.1.1 The quotient map P(Rn ) → P(Rn )/ Im DivωA ∼ =R is the map that sends a function f to its expected value n f ωA . f A := R Rn ωA This lemma plays a crucial motivational role for us. If we want to know expected values (which are the main interest in the physics setting), it suffices to describe a divergence operator. One does not need to produce the measure directly. One nice feature of this approach to finite-dimensional Gaussian integrals is that it works over any ring in which det A is invertible (this follows from the explicit algebraic formula we wrote for the divergence of a polynomial vector

2.2 Divergence in Infinite Dimensions

27

field). This way of looking at finite-dimensional Gaussian integrals was analyzed further in Gwilliam and Johnson-Freyd (2012), where it was shown that one can derive the Feynman rules for finite-dimensional Gaussian integration from such considerations. Remark: We should acknowledge here that our choice of polynomial functions and vector fields was important. Polynomial functions are integrable against a Gaussian measure, and the divergence of polynomial vector fields produces all the relations between these integrands. If we worked with all smooth functions and smooth vector fields, the cokernel would be zero. In the BV formalism, just as in ordinary integration, the choice of functions plays an important role. ♦

2.2 Divergence in Infinite Dimensions So far, we have seen that finite-dimensional Gaussian integrals are entirely encoded in the divergence map from the Gaussian measure. In our approach to infinite-dimensional Gaussian integrals, the fundamental object we will define is such a divergence operator. We will recover the usual formulae for infinitedimensional Gaussian integrals (in terms of the propagator or Green’s function) from our divergence operator. Further, we will see that analyzing the cokernel of the divergence operator will lead naturally to the notion of prefactorization algebra. For concreteness, we will work with the free scalar field theory on a Riemannian manifold (M, g), which need not be compact. We will define a divergence operator for the putative Gaussian measure on C∞ (M) associated to the quadratic form 12 M φ(g + m2 )φ dvolg . (Here g + m2 plays the role that the matrix A did in the preceding section.) Before we define the divergence operator, we need to define spaces of polynomial functions and of polynomial vector fields. We will organize these spaces by their support in M. Namely, for each open subset U ⊂ M, we will define polynomial functions and vector fields on the space C∞ (U). The space of all continuous linear functionals on C∞ (U) is the space Dc (U) of compactly supported distributions on U. To define the divergence operator, we need to restrict to functionals with more regularity. Hence we will work with Cc∞ (U), where every element f ∈ Cc∞ (U) defines a linear functional on C∞ (U) by the formula  f φ dvolg .

φ → U

ym (People sometimes call them “smeared” because these do not include beloved functionals such as delta functions, only smoothed-out approximations to them.)

28

From Gaussian Measures to Factorization Algebras

As a first approximation to the algebra we wish to use, we define the space of polynomial functions on Cc∞ (U) to be  P(C∞ (U)) = Sym Cc∞ (U), P(Cc∞ (U)) that is homoi.e., the symmetric algebra on Cc∞ (U). An element of  geneous of degree n can be written as a finite sum of monomials f1 · · · fn where the fi ∈ Cc∞ (U). Such a monomial defines a function on the space C∞ (U) of fields by the formula  f1 (x1 )φ(x1 ) . . . fn (xn )φ(xn ) dvolg (x1 ) ∧ · · · ∧ dvolg (xn ). φ → (x1 ,...,xn )∈U n

Note that because Cc∞ (U) is a topological vector space, it is more natural to use an appropriate completion of this purely algebraic symmetric power Symn Cc∞ (U). Because this version of the algebra of polynomial functions is a little less natural than the completed version, which we will introduce shortly, we use the notation  P. The completed version is denoted P. We define the space of polynomial vector fields in a similar way. Recall that if V is a finite-dimensional vector space, then the space of polynomial vector fields on V is isomorphic toP(V) ⊗ V, where P(V) is the space of polynomial functions on V. An element X = f ⊗ v, with f a polynomial, acts on a polynomial g by the formula X(g) = f

∂g . ∂v

In particular, if g is homogeneous of degree n and we pick a representative  g ∈ (V ∗ )⊗n , then ∂g/∂v denotes the degree n − 1 polynomial w →  g(v ⊗ w ⊗ · · · ⊗ w). In other words, for polynomials, differentiation is a version of contraction. In the same way, we would expect to work with

∞ (U)) =  P(C∞ (U)) ⊗ C∞ (U). Vect(C We are interested, in fact, in a different class of vector fields. The space C∞ (U) has a foliation, coming from the linear subspace Cc∞ (U) ⊂ C∞ (U). We are actually interested in vector fields along this foliation, owing to the role of variational calculus in field theory. This restriction along the foliation is clearest in terms of the divergence operator we describe later, so we explain it after Definition 2.2.1. Thus, let

c (C∞ (U)) =  P(C∞ (U)) ⊗ Cc∞ (U). Vect

2.2 Divergence in Infinite Dimensions

29

Again, it is more natural to use a completion of this space that takes account of the topology on Cc∞ (U). We will discuss such completions shortly.

c (C∞ (U)) can be written as an finite sum of monomials Any element of Vect of the form ∂ f1 · · · fn ∂φ ∂ for fi , φ ∈ Cc∞ (U). By ∂φ we mean the constant-coefficient vector field given by infinitesimal translation in the direction φ in C∞ (U). Vector fields act on functions, in the same way as we described earlier: the formula is   ∂ gi (x)φ(x) dvolg g1 · · · gi · · · gm f1 · · · fn (g1 · · · gm ) = f1 · · · fn ∂φ U

where dvolg is the Riemannian volume form on U. Definition 2.2.1 The divergence operator associated to the quadratic form  S(φ) = φ( + m2 )φ dvolg U

is the linear map

: Vect

c (C∞ (U)) →  Div P(Cc∞ (U)) defined by 

 n  ∂ 2 

= −f1 · · · fn (+m )φ+ Div f1 · · · fn f1 · · · fi · · · fn φ(x)fi (x) dvolg . ∂φ U i=1 (‡) Note that this formula is entirely parallel to the formula for divergence of a Gaussian measure in finite dimensions, given in formula (†). Indeed, the formula makes sense even when φ is not compactly supported; however, the term f1 · · · fn ( + m2 )φ need not be compactly supported if φ is not compactly supported. To ensure that the image of the divergence operator is in  P(Cc∞ (U)), we work only with

c (C∞ (U)). vector fields with compact support, namely Vect As we mentioned previously, it is more natural to use a completion of the spaces  P(C∞ (U)) and Vectc (C∞ (U)) of polynomial functions and polynomial vector fields. We now explain a geometric approach to such a completion. Let dvolgn denote the Riemannian volume form on the product space U n arising from the natural n-fold product metric induced by the metric g on U.

30

From Gaussian Measures to Factorization Algebras

Any element F ∈ Cc∞ (U n ) then defines a polynomial function on C∞ (U) by  φ → F(x1 , . . . , xn )φ(x1 ) · · · φ(xn ) dvolgn . Un

This functional does not change if we permute the arguments of F by an element of the symmetric group Sn , so that this function depends only on the image of F in the coinvariants of Cc∞ (U n ) by the symmetric group action. This quotient, of course, is isomorphic to invariants for the symmetric group action. Therefore we define  Cc∞ (U n )Sn , P(C∞ (U)) = n≥0

where the subscript indicates coinvariants. The (purely algebraic) symmetric power Symn Cc∞ (U) provides a dense subspace of Cc∞ (U n )Sn ∼ = Cc∞ (U n )Sn . ∞ ∞  Thus, P(C (U)) is a dense subspace of P(C (U)). In a similar way, we define Vectc (C∞ (U)) by  Cc∞ (U n+1 )Sn , Vectc (C∞ (U)) = n≥0

where the symmetric group Sn acts only on the first n factors. A dense subspace

c (C∞ (U)) is of Cc∞ (U n+1 )Sn is given by Symn Cc∞ (U) ⊗ Cc∞ (U) so that Vect ∞ a dense subspace of Vectc (C (U)). Lemma 2.2.2 The divergence map

: Vect

c (C∞ (U)) →  Div P(C∞ (U)) extends continuously to a map Div : Vectc (C∞ (U)) → P(C∞ (U)). Proof Suppose that F(x1 , . . . , xn+1 ) ∈ Cc∞ (U n+1 )Sn ⊂ Vectc (C∞ (U)). The divergence map in equation (‡) extends to a map that sends F to n   F(x1 , . . . , xi , . . . , xn , xi ) dvolg . −xn+1 F(x1 , . . . , xn+1 ) + i=1

xi ∈U

Here, xn+1 denotes the Laplacian acting only on the n + 1st copy of U. Note that the integral produces a function on U n−1 . With these objects in hand, we are able to define the quantum observables of a free field theory.

2.3 The Prefactorization Structure on Observables

31

Definition 2.2.3 For an open subset U ⊂ M, let H 0 (Obsq (U)) = P(C∞ (U))/ Im Div . In other words, H 0 (Obsq (U)) be the cokernel of the operator Div. Later we will see that this linear map Div naturally extends to a cochain complex of quantum observables, which we will denote Obsq , whose zeroth cohomology is what we just defined. This extension is why we write H 0 . Let us explain why we should interpret this space as the quantum observables. We expect that an observable in a field theory is a function on the space of fields. An observable on a field theory on an open subset U ⊂ M is a function on the fields that depends only on the behavior of the fields inside U. Speaking conceptually, the expectation value of the observable is the integral of this function against the “functional measure” on the space of fields. Our approach is that we do not try to define the functional measure, but instead we define the divergence operator. If we have some functional on C∞ (U) that is the divergence of a vector field, then the expectation value of the corresponding observable is zero. Thus, we would expect that the observable given by a divergence is not a physically interesting quantity, since its value is zero. Thus, we might as well identify it with zero. The appropriate vector fields on C∞ (U) – the ones for which our divergence operator makes sense – are vector fields along the foliation of C∞ (U) by compactly supported fields. Thus, the quotient of functions on C∞ (U) by the subspace of divergences of such vector fields gives a definition of observables.

2.3 The Prefactorization Structure on Observables Suppose that we have a Gaussian measure ωA on Rn . Then every function on Rn with polynomial growth is integrable, and this space of functions forms a commutative algebra. We showed that there is a short exact sequence DivωA

EωA

Vect(Rn ) −→ P(Rn ) −→ R → 0, where EωA denotes the expectation value map for this measure. But the image of the divergence operator is not an ideal. (Indeed, usually an expectation value map is not an algebra map!) This fact suggests that, in the BV formalism, the quantum observables should not form a commutative algebra. One can check quickly that for our definition given earlier, H 0 (Obsq (U)) is not an algebra. However, we will find that some shadow of this commutative algebra structure exists, which allows us to combine observables on disjoint subsets. This

32

From Gaussian Measures to Factorization Algebras

residual structure will give the spaces H 0 (Obsq (U)) of observables, viewed as a functor on the category of open subsets U ⊂ M, the structure of a prefactorization algebra. Let us make these statements precise. Note that P(C∞ (U)) is a commutative algebra, as it is a space of polynomial functions on C∞ (U). Further, if U ⊂ V there is a map of commutative algebras ext : P(C∞ (U)) → P(C∞ (V)), extending a polynomial map F : C∞ (U) → R to the polynomial map F ◦ res : C∞ (V) → R by precomposing with the restriction map res : C∞ (V) → C∞ (U). This map ext is injective. We will sometimes refer to an element of the subspace P(C∞ (U)) ⊂ P(C∞ (V)) as an element of P(C∞ (V)) with support in U. Lemma 2.3.1 The product map P(C∞ (V)) ⊗ P(C∞ (V)) → P(C∞ (V)) does not descend to a map H 0 (Obs(V)) ⊗ H 0 (Obs(V)) → H 0 (Obs(V)). If U1 , U2 are disjoint open subsets of the open V ⊂ M, then we have a map P(U1 ) ⊗ P(U2 ) → P(V) obtained by combining the inclusion maps P(Ui ) → P(V) with the product map on P(V). This map does descend to a map H 0 (Obs(U1 )) ⊗ H 0 (Obs(U2 )) → H 0 (Obs(V)). In other words, although the product of general observables does not make sense, the product of observables with disjoint support does. Proof Let U1 , U2 be disjoint open subsets of M, both contained in an open V. Let us view the spaces Vectc (C∞ (Ui )) and P(C∞ (Ui )) as subspaces of Vectc (C∞ (V)) and P(C∞ (V)), respectively. We denote by DivUi , the divergence operator on Ui , namely, DivUi : Vectc (C∞ (Ui )) → P(C∞ (Ui )). We use DivV to denote the divergence operator on V. Our situation is then described by the following diagram: ker q12

→ P(U1 ) ⊗ P(U2 ) ↓

Im DivV

→

P(C∞ (V))

−→

q12

H 0 (Obs(U1 )) ⊗ H 0 (Obs(U2 ))

q

H 0 (Obs(V)).

−→

2.3 The Prefactorization Structure on Observables

33

The middle vertical arrow is multiplication map. We want to show there is a vertical arrow on the right that makes a commuting square. It suffices to show that the image of ker q12 in H 0 (Obs(V)) is zero. Note that H 0 (Obs(U1 )) ⊗ H 0 (Obs(U2 )) is the cokernel of the map P(C∞ (U1 )) ⊗ Vectc (C∞ (U2 )) ⊕ Vectc (C∞ (U1 )) ⊗ P(C∞ (U2 )) 1⊗DivU2 + DivU1 ⊗1

−−−−−−−−−−−−→ P(C∞ (U1 )) ⊗ P(C∞ (U2 )).  Hence, ker q12 = Im 1 ⊗ DivU2 + DivU2 ⊗1 . We will show that the image of 1 ⊗ DivU2 + DivU1 ⊗1 sits inside Im DivV . This result ensures that we can produce the desired map. Thus, it suffices to show that for any F is in P(C∞ (U1 )) and X is in X in Vectc (C∞ (V)) such that DivV ( X) = F Vectc (C∞ (U2 )), there exists  DivU2 (X). (The same argument applies after switching the roles U1 and U2 .) We will show, in fact, that Div(FX) = F Div(X), viewing F and X as living on the open V. A priori, this assertion should be surprising. On an ordinary finitedimensional manifold, the divergence Div with respect to any volume form has the following property: for any vector field X and any function f , we have Div(fX) − f Div X = X(f ), where X(f ) denotes the action of X on f . Note that X(f ) is not necessarily in the image of Div, in which case f Div X is not in the image of Div, and so we see that Im Div is not an ideal. The same equation holds for the divergence operator we have defined in infinite dimensions. If X ∈ Vectc (C∞ (V)) and F ∈ P(Cc∞ (V)), then Div(FX) − F DivX = X(F). This computation tells us that the image of Div is not an ideal, as there exist X(F) not in the image of Div. When F is in P(C∞ (U1 )) and X is in Vectc (C∞ (U2 )), however, X(F) = 0, as their supports are disjoint. Thus, we have precisely the desired relation F Div(X) = Div(FX). We now prove that X(F) = 0 when X and F have disjoint support. We are working with polynomial functions and vector fields, so that all computations can be done in a purely algebraic fashion; in other words, we will work with derivations as in algebraic geometry. Let ε satisfy ε2 = 0. For a polynomial vector field X and a polynomial function F on a vector space V,

34

From Gaussian Measures to Factorization Algebras

we define the function X(F) to assign to the vector v ∈ V, the ε component of F(v + εXv ) − F(v). Here Xv denotes the tangent vector at v that X produces. In our situation, we know that for any φ ∈ C∞ (V), we have the following properties: • F(φ) depends only on the restriction of φ to U1 . • Xφ is a function with support in U2 and hence vanishes away from U2 . Thus, F(φ + εX(φ)) = F(φ) as the restriction of φ + εX(φ) to U1 agrees with the restriction of φ. We see then that X(F)(φ) =

d  F(φ + εXφ ) − F(φ) = 0, dε

as asserted. In a similar way, if U1 , . . . , Un are disjoint opens all contained in V, then there is a map H 0 (Obsq (U1 )) ⊗ · · · ⊗ H 0 (Obsq (Un )) → H 0 (Obsq (V)) descending from the map P(U1 ) ⊗ · · · ⊗ P(Un ) → P(V) given by inclusion followed by multiplication. Thus, we see that the spaces H 0 (Obsq (U)) for open sets U ⊂ M are naturally equipped with the structure maps necessary to define a prefactorization algebra. (See Section 1.2 in Chapter 1 for a sketch of the definition of a factorization algebra, and Section 3.1 in Chapter 3 for more details on the definition). It is straightforward to check, using the arguments from the preceding proof, that these structure maps satisfy the necessary compatibility conditions to define a prefactorization algebra.

2.4 From Quantum to Classical Our general philosophy is that the quantum observables of a field theory are a factorization algebra given by deforming the classical observables. The classical observables are defined to be functions on the space of solutions to the equations of motion. We now examine how our construction can be seen as just such a deformation. Let us see first why this holds for a class of measures on finite dimensional vector spaces. Let S be a polynomial function on Rn . Let dn x denote

2.4 From Quantum to Classical

35

the Lebesgue measure on Rn , and consider the measure ω = e−S/h¯ dn x, where h¯ is a small parameter. The divergence with respect to ω is given by the formula

  ∂ 1  ∂S  ∂fi + . fi =− Divω fi ∂xi ∂xi ∂xi h¯ As before, let P(Rn ) denote the space of polynomial functions on Rn and let Vect(Rn ) denote the space of polynomial vector fields. The divergence operator Divω is a linear map map Vect(Rn ) → P(Rn ). Note that the operators Divω and h¯ Divω have the same image so long as h¯ = 0. When h¯ = 0, the operator h¯ Divω becomes the operator  ∂  ∂S fi → − fi . ∂xi ∂xi Therefore, the h¯ → 0 limit of the image of Divω is the Jacobian ideal

 ∂S ⊂ P(Rn ), Jac(S) = ∂xi which corresponds to the critical locus of S in Rn . Hence, the h¯ → 0 limit of the observables P(Rn )/ Im Divω is the commutative algebra P(Rn )/ Jac(S) that describes functions on the critical locus of S. Let us now check the analogous property for the observables of a free scalar field theory on a manifold M. We will consider the divergence for the putative Gaussian measure 

 2 1 φ( + m )φ dφ exp − h¯ M

C∞ (M).

on map

For any open subset U ⊂ M, this divergence operator gives us a Divh¯ : Vectc (C∞ (U)) → P(C∞ (U))

with f1 · · · f n

 ∂ → − 1h¯ f1 · · · fn ( + m2 )φ + f1 · · ·  fi ∂φ i

 f φ. M

Note how h¯ appears in this formula; it is just the same modification of the operator (‡) as Divω is of the divergence operator for the measure e−S dn x. As in the finite dimensional case, the first term dominates in the h¯ → 0 limit. The h¯ → 0 limit of the image of Divh¯ is the closed subspace of P(C∞ (U)) spanned by functionals of the form f1 · · · fn ( + m2 )φ, where fi and φ are in Cc∞ (U)). This subspace is the topological ideal in P(C∞ (U)) generated by linear functionals of the form ( + m2 )f , where f ∈ Cc∞ (U). If S(φ) =

36

From Gaussian Measures to Factorization Algebras



φ( + m2 )φ is the action functional of our theory, then this subspace is precisely the topological ideal generated by all the functional derivatives   ∂S : φ ∈ Cc∞ (U) . ∂φ

In other words, it is the Jacobian ideal for S and hence is cut out by the Euler– Lagrange equations. Let us call this ideal IEL (U). A more precise statement of what we have just sketched is the following. Define a prefactorization algebra H 0 (Obscl (U)) (the superscript cl stands for classical) that assigns to U the quotient algebra P(C∞ (U))/IEL (U). Thus, H 0 (Obscl (U)) should be thought of as the polynomial functions on the space of solutions to the Euler–Lagrange equations. Note that each constituent space H 0 (Obscl (U)) in this prefactorization algebra has the structure of a commutative algebra, and the structure maps are all maps of commutative algebras. In short, H 0 (Obscl ) forms a commutative prefactorization algebra. Heuristically, this terminology means that the product map defining the factorization structure is defined for all pairs of opens U1 , U2 ⊂ V, and not just disjoint pairs. Our work in the section is summarized as follows. q

Lemma 2.4.1 There is a prefactorization algebra H 0 (Obsh¯ ) over C[h] ¯ such q cl 0 0 that when specialized to h¯ = 1 is H (Obs ) and to h¯ = 0 is H (Obs ). This prefactorization algebra assigns to an open set the cokernel of the map ∞ h¯ Divh¯ : Vectc (C∞ (U))[h] ¯ → P(C (U))[h], ¯

where Divh¯ is the map defined previously. q although this We will see later that H 0 (Obsh¯ (U)) is free as an R[h]-module, ¯ is a special property of free theories and is not always true for an interacting theory.

2.5 Correlation Functions We have seen that the observables of a free scalar field theory on a manifold M give rise to a factorization algebra. In this section, we explain how the structure of a factorization algebra is enough to define correlation functions of observables. We will calculate certain correlation functions explicitly and recover the standard answers.

2.5 Correlation Functions

37

Suppose now that M is a compact Riemannian manifold, and, as before, let us consider the observables of the free scalar field theory on M with mass m > 0. Then we have the following result. Lemma 2.5.1 If the mass m is positive, then H 0 (Obsq (M)) ∼ = R. Compare this result with the statement that for a Gaussian measure on Rn , the image of the divergence map is of codimension 1 in the space of polynomial functions on Rn . The assumption here that the mass is positive is necessary to ensure that the quadratic form M φ( + m2 )φ is nondegenerate. This lemma will follow from our more detailed analysis of free theories in Chapter 4, but we can sketch the idea here. The main point is that there is a q cl family of operators over R[h] ¯ connecting H 0 (Obs (M)) to H 0 (Obs (M)). It cl 0 is straightforward to see that the algebra H (Obs (M)) is R. Indeed, observe that the only solution to the equations of motion in the case that m > 0 is the function φ = 0, since the assumption that  has nonnegative spectrum ensures that φ = −m2 φ has no nontrivial solution. Functions on a point are precisely R. To conclude that H 0 (Obsq (M)) is also R, we need to show that q H 0 (Obsh¯ (M)) is flat over R[h], ¯ which will follow from a spectral sequence computation we will perform later in the book. There is always a canonical observable 1 ∈ H 0 (Obsq (U)) for any open subset U ⊂ M. This element is defined to be the image of the function 1 ∈ P(C∞ (U)). We identify H 0 (Obsq (M)) with R by taking this observable 1 to be a basis vector. Definition 2.5.2 Let U1 , . . . , Un ⊂ M be disjoint open subsets. The correlator is the prefactorization structure map − : H 0 (Obsq (U1 )) ⊗ · · · ⊗ H 0 (Obsq (Un )) → H 0 (Obsq (M)) ∼ = R. We should compare this definition with what happens in finite dimensions. If we have a Gaussian measure on Rn , then the space of polynomial functions modulo divergences is one-dimensional. If we take the image of a function 1 to be a basis of this space, then we get a map P(Rn ) → R from polynomial functions to R. This map is the integral against the Gaussian measure, normalized so that the integral of the function 1 is 1. In our infinite dimensional situation, we are doing something very similar. Any reasonable definition of the correlation function of the observables O1 , . . . , On , with Oi in P(C∞ (Ui )), should depend only on the product function O1 · · · On ∈ P(C∞ (M)). Thus, the correlation function map should be a

38

From Gaussian Measures to Factorization Algebras

linear map P(C∞ (M)) → R. Further, it should send divergences to zero. We have seen that there is only one such map, up to an overall scale.

2.5.1 Comparison to Physics Next we will check explicitly that this correlation function map really matches up with what physicists expect. Let fi ∈ Cc∞ (Ui ) be compactly supported smooth functions on the open sets U1 , U2 ⊂ M. Let us view each fi as a linear function on C∞ (Ui ), and so as an element of the polynomial functions P(C∞ (Ui )). Let G ∈ D(M × M) be the unique distribution on M × M with the property that (x + m2 )G(x, y) = δDiag . Here δDiag denotes the delta function supported on the diagonal copy of M inside M × M. In other words, if we apply the operator  + m2 to the first factor of G, we find the delta function on the diagonal. Thus, G is the kernel for the operator ( + m2 )−1 . In the physics literature, G is called the propagator; in the mathematics literature, it is called the Green’s function for the operator  + m2 . Note that G is smooth away from the diagonal. Lemma 2.5.3 Given fi ∈ Cc∞ (Ui ), there are classes [fi ] ∈ H 0 (Obsq (Ui )). Then  [f1 ] [f2 ] = f1 (x)G(x, y)f2 (y). x,y∈M

Note that this result is exactly the standard result in physics. Proof Later we will give a slicker and more general proof of this kind of statement. Here we will give a simple proof to illustrate how the Green’s function arises from our homological approach to defining functional integrals. The operator  + m2 is surjective. Thus, there is a preimage φ = ( + m2 )−1 f2 ∈ Cc∞ (M), that is unique because  + m2 is also injective. Indeed, we know that  φ(x) = G(x, y)f2 (y). y∈M

Now consider the vector field f1

∂ ∈ Vectc (C∞ (M)). ∂φ

2.6 Further Results on Free Field Theories

Note that



∂ Div f1 ∂φ



39



  f1 (x)φ(x) − f1 ( + m2 )φ dvol x∈M 

 f1 (x)G(x, y)f2 (y) · 1 − f1 f2 . =

=

M

P(C∞ (M)) is a cocycle representing the factorization prod-

The element f1 f2 ∈ uct [f1 ][f2 ] in H 0 (Obsq (M)) of the observables [fi ] ∈ H 0 (Obsq (Ui )). The displayed equation tells us that

  f1 (x)G(x, y)f2 (y) · 1 ∈ H 0 (Obsq (M)). [f1 f2 ] = M

Since the observable 1 is chosen to be the basis element identifying H 0 (Obsq (M)) with R, the result follows. With a little more work, the same arguments recover the usual Wick’s formula for correlators of the form f1 · · · fn . Remark: These kinds of formulas are standard knowledge in physics, but not in mathematics. For a more extensive discussion in a mathematical style, see chapter 6 of Glimm and Jaffe (1987) or lecture 3 by Kazhdan in Deligne et al. (1999). ♦

2.6 Further Results on Free Field Theories In this chapter, we showed that if we define the observables of a free field theory as the cokernel of a certain divergence operator, then these spaces of observables form a prefactorization algebra. We also showed that this prefactorization algebra contains enough information to allow us to define the correlation functions of observables, and that for linear observables we find the same formula that physicists would write. In Chapter 3 we show that a certain class of factorization algebras on the real line are equivalent to associative algebras, together with a derivation. As Noether’s theorem suggests, the derivation arises from infinitesimal translation on the real line, so that it encodes the Hamiltonian of the system. In Chapter 4, we analyze the factorization algebra of free field theories in more detail. We will show that if we consider the free field theory on R, the factorization algebra H 0 (Obsq ) corresponds (under the relationship between factorization algebras on R and associative algebras) to the Weyl algebra. The Weyl algebra is generated by observables p, q corresponding to position and

40

From Gaussian Measures to Factorization Algebras

momentum satisfying [p, q] = 1. If we consider instead the family over R[h] ¯ q of factorization algebras H 0 (Obsh¯ ) discussed earlier, then we find the commutation relation [p, q] = h¯ of Heisenberg. This algebra, of course, is what is traditionally called the algebra of observables of quantum mechanics. In this case, we will further see that the derivation of this algebra (corresponding to infinitesimal time translation) is an inner derivation, given by bracketing with the Hamiltonian H = p2 − m2 q2 , which is the standard Hamiltonian for the quantum mechanics of the harmonic oscillator. More generally, we recover canonical quantization of the free scalar field theory on higher dimensional manifolds as follows. Consider a free scalar theory on the product Riemannian manifold N × R, where N is a compact Riemannian manifold. This example gives rise to a factorization algebra on R that assigns to an open subset U ⊂ R, the space H 0 (Obsq (N × U)). We will see that this factorization algebra on R has a dense sub-factorization algebra corresponding to an associative algebra. This associative algebra is the tensor product of Weyl algebras, where each each eigenspace of the operator  + m2 on C∞ (N) produces a Weyl algebra. In other words, we find quantum mechanics on R with values in the infinite-dimensional vector space C∞ (N). Since that space has a natural spectral decomposition for the operator  + m2 , it is natural to interpret as the algebra of observables, the associative algebra given by tensoring together the Weyl algebra for each eigenspace. This result is entirely consistent with standard arguments in physics, being the factorization algebra analog of canonical quantization.

2.7 Interacting Theories In any approach to quantum field theory, free field theories are easy to construct. The challenge is always to construct interacting theories. The core results of this two-volume work show how to construct the factorization algebra corresponding to interacting field theories, deforming the factorization algebra for free field theories discussed earlier. Let us explain a little bit about the challenges we need to overcome to deal with interacting theories, and how we overcome these challenges.

2.7 Interacting Theories

41

Consider an interacting scalar field theory on a Riemannian manifold M. For instance, we could consider an action functional of the form   2 1 S(φ) = − 2 φ( + m )φ + φ4. M

M

In general, the action functional must be local: it must arise as the integral over M of some polynomial in φ and its derivatives. This condition is due to our interest in field theories, but it is also necessary to produce factorization algebras. We will let I(φ) denote the interaction term in our field theory, which consists of the cubic and higher terms in S. In the above example, I(φ) = φ 4 . We will always assume that the quadratic term in S is similar in form − 12 M φ(+ m2 )φ, i.e., we require an ellipticity condition. (Of course, the examples amenable to our techniques apply to a very general class of interacting theories, including many gauge theories.) If U ⊂ M is an open subset, we can consider, as before, the spaces Vectc (C∞ (M)) and P(C∞ (M)) of polynomial functions and vector fields on M. By analogy with the finite-dimensional situation, one can try to define the divergence for the putative measure exp S(φ)/hdμ (where dμ refers to the ¯ “Lebesgue measure” on C∞ (M))) by the formula

  ∂ ∂S  = 1h¯ f1 . . . fn + f1 . . .  fi . . . fn fi φ. Divh¯ f1 . . . fn ∂φ ∂φ M This formula agrees with the formula we used when S was purely quadratic. But now a problem arises. We defined P(C∞ (U)) as the space of polynomial functions whose Taylor terms are given by integration against a smooth function on U n . That is,  P(C∞ (U)) = Cc∞ (U n )Sn . n ∂S is not necessarily in this space of Unfortunately, if φ ∈ Cc∞ (U)), then ∂φ 4 functions. For instance, if I(φ) = φ is the interaction term in the example of the φ 4 theory, then  ∂I (ψ) = φψ 3 dvol. ∂φ M ∂I provides a cubic function on the space C∞ (U), but it is not given by Thus ∂φ integration against an element in Cc∞ (U 3 ). Instead it is given by integrating against a distribution on U 3 , namely, the delta distribution on the diagonal.

42

From Gaussian Measures to Factorization Algebras

We can try to solve this issue by using a larger class of polynomial functions. Thus, we could let  P(C∞ (U)) = Dc (U n )Sn n

(U n )

where Dc is the space of compactly supported distributions on U. Similarly, we could let  Vectc (U) = Dc (U n+1 )Sn . n

The spaces P(C∞ (U)) and Vectc (C∞ (U)) are dense subspaces of these spaces. If φ0 ∈ Dc (U) is a compactly supported distribution, then for any local ∂S is a well-defined element of P(C∞ (U)). Thus, it looks like we functional S, ∂φ have resolved our problem. However, using this larger space of functions gives us a new problem: the second term in the divergence operator now fails to be well defined! For example, if f , φ ∈ Dc (U), then we have

  ∂ 1 ∂S + f φ. Divh¯ f = h¯ f ∂φ ∂φ M Now, M f φ no longer make sense, because we are trying to multiply distributions. More explicitly, this term involves pairing the distribution f  φ on the diagonal in M 2 with the delta function on the diagonal. If we consider the h¯ → 0 limit of this putative operator h¯ Divh¯ , we nonetheless find a well-defined operator Vectc (C∞ (U)) → P(C∞ (U)) X → XS, which sends a vector field X to its action on the local functional S. The cokernel of this operator is the quotient of P(C∞ (U)) by the ideal IEL (U) generated by the Euler–Lagrange equations. We thus let cl

H 0 (ObsS (U)) = P(C∞ (U))/IEL . As U varies, this construction produces a factorization algebra on M, which we call the factorization algebra of classical observables associated to the action functional S. Lemma 2.7.1 If S = − 12 φ( + m2 )φ, then this definition of classical observables coincides with the one we discussed earlier: H 0 (ObsS (U)) ∼ = H 0 (Obscl (U)) cl

where H 0 (Obscl (U)) is defined, as earlier, to be the quotient of P(C∞ (U)) by the ideal IEL (U) of the Euler–Lagrange equations.

2.7 Interacting Theories

43

This result is a version of elliptic regularity, and we prove it later, in Chapter 4. Now the challenge we face should be clear. If S is the action functional for the free field theory, then we have a factorization algebra of classical observables. This factorization algebra deforms in two ways. First, we can deform it into the factorization algebra of quantum observables for a free theory. Second, we can deform it into the factorization algebra of classical observables for an interacting field theory. The difficulty is to perform both of these deformations simultaneously. To construct the observables of an interacting field theory, we use the renormalization technique of Costello (2011b). In Costello (2011b), the first author gives a definition of a quantum field theory and a cohomological method for constructing field theories. A field theory as defined in Costello (2011b) gives us (essentially from the definition) a family of divergence operators Div[L] : Vectc (M) → P(C∞ (M)), one for every L > 0. These divergence operators, for varying L, can be conjugated to each other by continuous linear isomorphisms of Vectc (M)) and P(C∞ (M)). However, for a proper open subset U, these divergence operators do not map Vectc (U)) into P(C∞ (U)). However, roughly speaking, for L very small, the operator Div[L] only increases the support of a vector field in Vectc (U)) by a small amount outside of U. For small enough L, it is almost support preserving. This property turns out to be enough to define the factorization algebra of quantum observables. This construction of quantum observables for an interacting field theory is given in Volume 2, which is the most technically difficult part of this work. Before tackling it, we will develop more language and explore more examples. In particular, we will • Develop some formal and structural aspects of the theory of factorization algebras. • Analyze in more detail the factorization algebra associated to a free field theory. • Construct and analyze factorization algebras associated to vertex algebras such as the Kac–Moody vertex algebra. • Develop classical field theory using a homological approach arising from the BV formalism. • Flesh out the description of the factorization algebra of classical observables we have sketched here. The example of this chapter, however, already exhibits the central ideas.

3 Prefactorization Algebras and Basic Examples

In this chapter we give a formal definition of the notion of prefactorization algebra. With the definition in hand, we proceed to examine several examples that arise naturally in mathematics. In particular, we explain how associative algebras can be viewed as prefactorization algebras on the real line, and when the converse holds. We also explain how to construct a prefactorization algebra from a sheaf of Lie algebras on a manifold M. This construction is called the factorization envelope, and it is related to the universal enveloping algebra of a Lie algebra as well as to Beilinson–Drinfeld’s notion of a chiral envelope. Although the factorization envelope construction is very simple, it plays an important role in field theory. For example, the factorization algebra for any free theories is a factorization envelope, as is the factorization algebra corresponding to the Kac– Moody vertex algebra. More generally, factorization envelopes play an important role in our formulation of Noether’s theorem for quantum field theories. Finally, when the manifold M is equipped with an action of a group G, we describe what a G-equivariant prefactorization algebra is. We will use this notion later in studying translation-invariant field theories (see Section 4.8 in Chapter 4) and holomorphically translation-invariant field theories (see Chapter 5).

3.1 Prefactorization Algebras In this section we give a formal definition of the notion of a prefactorization algebra, starting concretely and then generalizing. In the first subsection, using plain language, we describe a prefactorization algebra taking values in vector spaces. Readers are free to generalize by replacing “vector space” and “linear map” with “object of a symmetric monoidal category C” and “morphism in 44

3.1 Prefactorization Algebras

45

C.” (Our favorite target category is cochain complexes.) The next subsections give a concise definition using the language of multicategories (also known as colored operads) and allow an arbitrary multicategory as the target. In the final subsections, we describe the category (and multicategory) of such prefactorization algebras.

3.1.1 The Definition in Explicit Terms Let M be a topological space. A prefactorization algebra F on M, taking values in vector spaces, is a rule that assigns a vector space F(U) to each open set U ⊂ M along with the following maps and compatibilities. • There is a linear map mU V : F(U) → F(V) for each inclusion U ⊂ V. 1 ,...,Un • There is a linear map mU : F(U1 ) ⊗ · · · ⊗ F(Un ) → F(V) for every V finite collection of open sets where each Ui ⊂ V and the Ui are pairwise disjoint. The following picture represents the situation.

V U1 U2

1 ,...,Un : F(U1 ) ⊗ · · · ⊗ F(Un ) → F(V)  mU V

...

Un

• The maps are compatible in the obvious way, so that if Ui,1  · · ·  Ui,ni ⊆ Vi and V1  · · ·  Vk ⊆ W, the following diagram commutes. k ni i=1

k

j=1 F(Uj )

i=1 F(Vi )

F(W) Thus F resembles a precosheaf, except that we tensor the vector spaces rather than take their direct sum.

46

Prefactorization Algebras and Basic Examples



a⊗b⊗c

∈

A⊗A⊗A

ab ⊗ c

∈

A⊗A

abc

∈

A

Figure 3.1. The prefactorization algebra Afact of an associative algebra A.

For an explicit example of the associativity, consider the following picture.

V1 U1,2 U1,1

W F(U1,1 ) ⊗ F(U1,2 ) ⊗ F(U2,1 ) V2

 F(V1 ) ⊗ F(V2 )

U2,1

F(W)

The case of k = n1 = 2, n2 = 1 These axioms imply that F(∅) is a commutative algebra. We say that F is a unital prefactorization algebra if F(∅) is a unital commutative algebra. In this case, F(U) is a pointed vector space by the image of the unit 1 ∈ F(∅) under the structure map F(∅) → F(U). In practice, for our examples, F(∅) is C, R, C[[h]], ¯ or R[[h]]. ¯ Example: The crucial example to bear in mind is an associative algebra. Every associative algebra A defines a prefactorization algebra Afact on R, as follows. To each open interval (a, b), we set Afact ((a, b)) = A. To any open set U =   j Ij , where each Ij is an open interval, we set F(U) = j A. The structure maps simply arise from the multiplication map for A. Figure 3.1 displays the structure of Afact . Notice the resemblance to the notion of an E1 or A∞ algebra. (One takes an infinite tensor products of unital algebras, as follows. Given an infinite set I, consider the poset of finite subsets of I, ordered by inclusion. For  each finite subset J ⊂ I, we can take the tensor product AJ = j∈J A. For 

J → J  , we define a map AJ → AJ by tensoring with the identity 1 ∈ A for ♦ every j ∈ J  \J. Then AI is the colimit over this poset.)

3.1 Prefactorization Algebras

47

Example: Another important example for us is the symmetric algebra of a precosheaf. Let F be a precosheaf of vector spaces on a space X. (For example, consider F = Cc∞ the compactly supported smooth functions on a manifold.) The functor F = Sym F : U → Sym(F(U)) defines a precosheaf of commutative algebras, but it also a prefactorization algebra. For instance, let U and V be disjoint opens. The structure maps F(U) → F(U  V) and F(U) → F(U  V) induce a canonical map F(U) ⊕ F(V) → F(U  V), and so we obtain a natural map Sym(F(U) ⊕ F(V)) → Sym F(U  V). But F(U) ⊗ F(V) ∼ = Sym(F(U) ⊕ F(V)), = Sym(F(U)) ⊗ Sym(F(V)) ∼ so there is a natural map F(U) ⊗ F(V) → F(U  V). In a similar way, one can provide all the structure maps to make F a prefactorization algebra. ♦ In the remainder of this section, we describe two other ways of phrasing this idea, but the reader who is content with this definition and eager to see examples should feel free to jump ahead, referring back as needed.

3.1.2 Prefactorization Algebras as Algebras over an Operad We now provide a succinct and general definition of a prefactorization algebra using the efficient language of multicategories. (See Section A.2.3 in Appendix A for a quick overview of the notion of a multicategory, also known as a colored operad. Note that we mean the symmetric version of such definitions.) Definition 3.1.1 Let DisjM denote the following multicategory associated to M. • The objects consist of all connected open subsets of M. • For every (possibly empty) finite collection of open sets {Uα }α∈A and open set V, there is a set of maps DisjM ({Uα }α∈A | V). If the Uα are pairwise disjoint and all are contained in V, then the set of maps is a single point. Otherwise, the set of maps is empty. • The composition of maps is defined in the obvious way. A prefactorization algebra just is an algebra over this colored operad DisjM . Definition 3.1.2 Let C be a multicategory. A prefactorization algebra on M taking values in C is a functor (of multicategories) from DisjM to C.

48

Prefactorization Algebras and Basic Examples

Since symmetric monoidal categories are special kinds of multicategories, this definition makes sense for C any symmetric monoidal category. Remark: If C is a symmetric monoidal category under coproduct, then a precosheaf on M with values in C defines a prefactorization algebra valued in C. Hence, our definition broadens the idea of “inclusion of open sets leads to inclusion of sections” by allowing more general monoidal structures to “combine” the sections on disjoint open sets. ♦ Note that if F is any prefactorization algebra, then F(∅) is a commutative algebra object of C. Definition 3.1.3 We say a prefactorization algebra F is unital if the commutative algebra F(∅) is unital. Remark: There is an important variation on this definition where one weakens the requirement that the composition of structure maps holds “on the nose” and instead requires homotopy coherence. For example, given disjoint opens U1 and U2 contained in V, which is then contained in W, we do not require 1 ,U2 1 ,U2 = mVW ◦ mU but that there is a “homotopy” between these that mU W V maps. This kind of situation arises naturally whenever the target category is best viewed as an ∞-category, such as the category of cochain complexes. We do not develop here the formalism necessary to treat homotopy-coherent prefactorization algebras because our examples and constructions always satisfy the strictest version of composition. Readers interested in seeing this variant developed should see the treatment in Lurie (2016). (We remark that one typically has “strictification” results that ensure that a homotopy-coherent algebra over a colored operad can be replaced by a weakly equivalent strict algebra over a colored operad, so that working with strict algebras is sufficient for many purposes.) ♦

3.1.3 Prefactorization Algebras in the Style of Precosheaves Any multicategory C has an associated symmetric monoidal category SC, which is defined to be the universal symmetric monoidal category equipped with a functor of multicategories C → SC. We call it the symmetric monoidal envelope of C. Concretely, an object of SC is a formal tensor product a1 ⊗ · · · ⊗ an of objects of C. Morphisms in SC are characterized by the property that for any object b in C, the set of maps SC(a1 ⊗ · · · ⊗ an , b) in the symmetric monoidal category is exactly the set of maps C({a1 , . . . , an } | b) in the multicategory C.

3.1 Prefactorization Algebras

49

We can give an alternative definition of prefactorization algebra by working with the symmetric monoidal category S DisjM rather than the multicategory DisjM . Definition 3.1.4 Let S DisjM denote the following symmetric monoidal category. • An object of S DisjM is a formal finite sequence [V1 , . . . , Vm ] of opens Vi in M. • A morphism F : [V1 , . . . , Vm ] → [W1 , . . . , Wn ] consists of a surjective function φ : {1, . . . , m} → {1, . . . , n} and a morphism fj ∈ DisjM ({Vk }k∈φ −1 (j) |Wj ) for each 1 ≤ j ≤ n. • The symmetric monoidal structure on S DisjM is given by concatenation. The alternative definition of prefactorization algebra is as follows. It resembles the notion of a precosheaf (i.e., a functor out of some category of opens) with the extra condition that it is symmetric monoidal. Definition 3.1.5 A prefactorization algebra with values in a symmetric monoidal category C ⊗ is a symmetric monoidal functor S DisjM → C . Remark: Although “algebra” appears in its name, a prefactorization algebra only allows one to “multiply” elements that live on disjoint open sets. The category of prefactorization algebras (taking values in some fixed target category) has a symmetric monoidal product, so we can study commutative algebra objects in that category. As an example, we will consider the observables for a classical field theory. ♦

3.1.4 Morphisms and the Category Structure We now explain how prefactorization algebras form a category. Definition 3.1.6 A morphism of prefactorization algebras φ : F → G consists of a map φU : F(U) → G(U) for each open U ⊂ M, compatible with the structure maps. That is, for any open V and any finite collection U1 , . . . , Uk of pairwise disjoint open sets, each contained in V, the following diagram commutes: F(U1 ) ⊗ · · · ⊗ F(Uk ) ↓ F(V)

φU1 ⊗···⊗φUk

−−−−−−−→ G(U1 ) ⊗ · · · ⊗ G(Uk ) . ↓ φV

−−−−−→

G(V)

Likewise, all the obvious associativity relations are respected.

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Prefactorization Algebras and Basic Examples

Remark: When our prefactorization algebras take values in cochain complexes, we require the φU to be cochain maps, i.e., they each have degree 0 and commute with the differentials. ♦ Definition 3.1.7 On a space X, we denote the category of prefactorization algebras on X taking values in the multicategory C by PreFA(X, C). Remark: When the target multicategory C is a model category or dg category or some other kind of higher category, the category of prefactorization algebras naturally forms a higher category as well. Given the nature of our constructions and examples in the next few chapters, such aspects do not play a prominent role. When we define factorization algebras in Chapter 6, however, we will discuss such issues. ♦

3.1.5 The Multicategory Structure Let SC denote the enveloping symmetric monoidal category of the multicategory C (see Section A.3.3 in Appendix A). Let ⊗ denote the symmetric monoidal product in SC. Any prefactorization algebra valued in C gives rise to one valued in SC. There is a natural tensor product on PreFA(X, SC), as follows. Let F, G be prefactorization algebras. We define F ⊗ G by (F ⊗ G)(U) = F(U) ⊗ G(U), and we simply define the structure maps as the tensor product of the structure maps. For instance, if U ⊂ V, then the structure map is U m(F)U V ⊗m(G)V : (F⊗G)(U) = F(U)⊗G(U) → F(V)⊗G(V) = (F⊗G)(V).

Definition 3.1.8 Let PreFAmc (X, C) denote the multicategory arising from the symmetric monoidal product on PreFA(X, SC). That is, if F1 , . . . Fn , G are prefactorization algebras valued in C, we define the set of multimorphisms to be PreFAmc (F1 , · · · , Fn | G) = PreFA(F1 ⊗ · · · ⊗ Fn , G), the set of maps of SC-valued prefactorization algebras from F1 ⊗· · ·⊗Fn to G.

3.2 Associative Algebras from Prefactorization Algebras on R

51

3.2 Associative Algebras from Prefactorization Algebras on R We explained in the preceding text how an associative algebra provides a prefactorization algebra on the real line. There are, however, prefactorization algebras on R that do not come from associative algebras. Here we characterize those that do arise from associative algebras. Definition 3.2.1 Let F be a prefactorization algebra on R taking values in the category of vector spaces (without any grading). We say F is locally constant if the map F(U) → F(V) is an isomorphism for every inclusion of intervals U ⊂ V. Lemma 3.2.2 Let F be a locally constant, unital prefactorization algebra on R taking values in vector spaces. Let A = F(R). Then A has a natural structure of an associative algebra. Remark: Recall that F being unital means that the commutative algebra F(∅) is equipped with a unit. We will find that A is an associative algebra over F(∅). ♦ Proof For any interval (a, b) ⊂ R, the map F((a, b)) → F(R) = A is an isomorphism. Thus, we have a canonical isomorphism A = F((a, b)) for all intervals (a, b). Notice that if (a, b) ⊂ (c, d) then the diagram A

∼ =

(a,b)

Id

 A

/ F((a, b))

∼ =

i(c,d)

 / F((c, d))

commutes. The product map m : A ⊗ A → A is defined as follows. Let a < b < c < d. Then, the prefactorization structure on F gives a map F((a, b)) ⊗ F((c, d)) → F((a, d)), and so, after identifying F((a, b)), F((c, d)) and F((a, d)) with A, we get a map A ⊗ A → A.

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Prefactorization Algebras and Basic Examples

This is the multiplication in our algebra. It remains to check the following. (i) This multiplication doesn’t depend on the intervals (a, b)(c, d) ⊂ (a, d) we chose, as long as (a, b) < (c, d). (ii) This multiplication is associative and unital. This is an easy (and instructive) exercise.

3.3 Modules as Defects We want to explain another simple but illuminating class of examples, and then we apply this perspective in the context of quantum mechanics. We will work with prefactorization algebras taking values in vector spaces with the tensor product as symmetric monoidal structure.

3.3.1 Modules as Living at Points We described already how to associate a prefactorization algebra FA on R to an associative algebra A. It is easy to modify this construction to describe bimodules, as follows. Let A and B be associative algebras and M an A-Bbimodule, i.e., M is a left A-module and a right B-module and these structures are compatible in that (am)b = a(mb) for all a ∈ A, b ∈ B, and m ∈ M. We now construct a prefactorization algebra FM on R that encodes this bimodule structure. Pick a point p ∈ R. On the half-line {x ∈ R | x < p}, FM is given by FA : to an interval I = (t0 , t1 ) with t1 < p, FM (I) = A, and the structure map for inclusion of finitely many disjoint intervals into a bigger interval is determined by multiplication in A. Likewise, on the half-line {x ∈ R | p < x}, FM is given by FB . Intervals containing p are determined, though, by M. When we consider an interval I = (t0 , t1 ) such that t0 < p < t1 , we set FM (I) = M. The structure maps are also determined by the bimodule structure. For example, given s0 < s1 < t0 < p < t1 < u0 < u1 , we have FM ((s0 , s1 )  (t0 , t1 )  (u0 , u1 )) = FM ((s0 , s1 )) ⊗ FM ((t0 , t1 )) ⊗ FM (u0 , u1 )) =A⊗M⊗B

3.3 Modules as Defects



53

a⊗m⊗b

∈

A⊗M⊗B

am ⊗ b

∈

M⊗B

amb

∈

M

Figure 3.2. The prefactorization algebra FM for the A-B-bimodule M.

and the inclusion of these three intervals into (s0 , u1 ) is the map FM ((s0 , s1 )) ⊗ FM ((t0 , t1 )) ⊗ FM )(u0 , u1 )) a⊗m⊗b

→ FM ((s0 , u1 )) . → amb

The definition of a bimodule ensures that we have a prefactorization algebra. See Figure 3.2. There is a structure map that we have not discussed yet, though. The inclusion of the empty set into an interval I containing p means that we need to pick an element mI of M for each interval. The simplest case is to fix one element m ∈ M and simply use it for every interval. If we are assigning the unit of A as the distinguished element for FM on every interval to the left of p and the unit of B for every interval to the right of p, then the distinguished elements, then the structure maps we have given clearly respect these distinguished elements. These distinguished elements, however, can change with the intervals, so long as they are preserved by the structure maps. For an interesting example, see the discussion of quantum mechanics that follows. Let us examine one more interesting case. Suppose we have algebras A, B, and C, and an A-B-bimodule M and a B-C-bimodule N. There is a prefactorization algebra on R describing the natural algebra for this situation. Fix points p < q. Let FM,N be the prefactorization algebra on R such that • On {x ∈ R | x < p}, it agrees with FA . • On {x ∈ R | p < x < q}, it agrees with FB . • On {x ∈ R | q < x}, it agrees with FC . and • To intervals (t0 , t1 ) with t0 < p < t1 < q, it assigns M. • To intervals (t0 , t1 ) with p < t0 < q < t1 , it assigns N. In other words, on {x ∈ R | x < q}, this prefactorization algebra FM,N behaves like FM , and like FN on {x ∈ R | p < x}.

54

Prefactorization Algebras and Basic Examples

We still need to describe what it does on an interval I of the form (T0 , T1 ) with T0 < p < q < T1 . There is a natural choice, dictated by the requirement that we produce a prefactorization algebra. We know that FM,N (I) must receive maps from M and N, by considering smaller intervals that contain either p or q, respectively. It also receives maps from A, B, and C from intervals not hitting these marked points, but these factor through intervals containing one of the marked points. Finally, we must have a structure map μ : M ⊗ N → FM,N (I) for each pair of disjoint intervals hitting both marked points. Note, in particular, what the associativity condition requires in the situation where we have three disjoint intervals given by s0 < p < s1 < t0 < t1 < u0 < q < u1 , contained in I. We can factor the inclusion of these three intervals through the pair of intervals (s0 , t1 )  (u0 , u1 ) or the pair of intervals (s0 , s1 )  (t0 , u1 ) Thus, our structure map M ⊗ B ⊗ N → FM,N (I) must satisfy that μ((mb) ⊗ n) = μ(m ⊗ (bn)) for every m ∈ M, b ∈ B, and n ∈ N. This condition means that FM,N (I) receives a canonical map from the tensor product M ⊗B N. Hence, the most natural choice is to set FM,N (I) = M ⊗B N. One can make other choices for how to extend to these longer intervals, but such a prefactorization algebra will receive a map from this one. The local-to-global principle satisfied by a factorization algebra is motivated by this kind of reasoning. Remark: We have shown how thinking about prefactorization algebras on a real line “decorated” with points (i.e., with a kind of stratification) reflects familiar algebraic objects like bimodules. By moving into higher dimensions and allowing more interesting submanifolds and stratifications, one generalizes this familiar algebra into new, largely unexplored directions. See Ayala et al. (2015) for an extensive development of these ideas in the setting of locally constant factorization algebras. ♦

3.3.2 Standard Quantum Mechanics as a Prefactorization Algebra We will now explain how to express the standard formalism of quantum mechanics in the language of prefactorization algebras, using the kind of construction just described. As our goal is to emphasize the formal structure, we will work

3.3 Modules as Defects

55

with a finite-dimensional complex Hilbert space and avoid discussions of functional analysis. Remark: In a sense, this section is a digression from the central theme of the book. Throughout this book we take the path integral formalism as fundamental, and hence we do not focus on the Hamiltonian, or operator, approach to quantum physics. Hopefully, juxtaposed with our work in Section 4.3 in Chapter 4, this example clarifies how to connect our methods with others. ♦ Let V denote a finite-dimensional complex Hilbert space V. That is, there is a nondegenerate symmetric sesquilinear form (−, −) : V × V → C so that (λv, λ v ) = λλ (v, v ) where λ, λ are complex numbers and v, v are vectors in V. Let A = End(V) T denote the algebra of endomorphisms, which has a ∗-structure via M ∗ = M , the conjugate-transpose. The space V is a representation of A, and the ∗-structure is characterized by the property that (M ∗ v, v ) = (v, Mv ). It should be clear that one could work more generally with a Hilbert space equipped with the action of a ∗-algebra of operators, aka observables. Now that we have fixed the kinematics of the situation, let’s turn to the dynamics. Let (Ut )t∈R be a one-parameter group of unitary operators on V. Since we are in the finite-dimensional setting, there is no problem identifying Ut = eitH for some Hermitian operator H that we call the Hamiltonian. We view V as a state space for our system, A as where the observables live, and H as determining the time evolution of our system. We now rephrase this structure to make it easier to articulate via the factorization picture. Let V denote V with the conjugate complex structure. We will denote elements of V by “kets” | v and elements of V by “bras” v | , and we provide a bilinear pairing between them by   v | v = (v, v ). We equip V with the right A-module structure by v | M = M ∗ v | .

  We will write v | M | v as can think of M acting on v from the left or on v from the right and it will produce the same number. Our goal is to describe a scattering-type experiment.

56

Prefactorization Algebras and Basic Examples

• At time t = 0, we prepare our system in the initial state vin | . • We modify the governing Hamiltonian over some finite time interval (i.e., apply an operator, or equivalently, an observable). • At time t = T, we measure whether our system is in the final state | vout . If we run this experiment many times, with the same initial and final states and the same operator, we should find a statistical pattern in our data. If an operator O acts during a time interval (t, t ), then we are trying to compute the number 

vin | eitH Oei(T−t )H | vout . Our formalism does not include the idealized situation of an operator O acting at a single moment t0 in time, but in the appropriate limit of shorter and shorter time intervals around t0 , we would compute vin | eit0 H Oei(T−t0 )H | vout , which agrees with the usual prescription. Note that we use a bra vin | for the “incoming” state and a ket | vout for the“outgoing” state so that the left-to-right ordering will agree with the left-to-right ordering of the real line viewed as a timeline. The prefactorization algebra F on the interval [0, T] describing this situation has the following structure. Interior open intervals describe moments when operators can act on our system. An interval that contains 0 (but not the other end) should describe possible “incoming” states of the system; dually, an interval containing the other endpoint should describe “outgoing” states. Let us now spell things out explicitly. To open subintervals, our prefactorization algebra F assigns the following vector spaces: • • • •

[0, t) → V (s, t) → A (t, T] → V [0, T] → C.

In light of our discussion about modules in the preceding section, note that the natural choice for the value F([0, T]) is the vector space V ⊗A V. However, V ⊗End(V) V is isomorphic to the ground field C due to the compatibility of the left and right actions with the inner product. We must now describe the structure maps coming from inclusion of intervals; we will describe enough so that the mechanism is clear. The case that determines the rest is that the inclusion [0, t0 )  (t1 , t2 )  (t3 , T] ⊂ [0, T]

3.3 Modules as Defects

57

corresponds to the structure map V ⊗A⊗V → v0 | ⊗ O ⊗ | v1 →

C . v0 | ei(t1 −t0 )H Oei(t3 −t2 )H | v1

In other words, the system evolves according the Hamiltonian during the closed intervals in the complement of opens during which we specify the incoming and outgoing states and the operator. Note that if we set O = ei(t2 −t1 )H , then we obtain v0 | ei(t1 −t3 )H | v1 and so recover the expected value of being in state v0 | at time t0 and going to state | v1 at time t3 . Setting t3 = t0 , we recover the inner product on V. For another example of structure maps, the inclusion (t0 , t1 ) ⊂ (t0 , t1 ) goes   to O → ei(t0 −t0 )H Oei(t1 −t1 )H . More generally, for k disjoint open intervals inside a big open interval (t0 , t1 )  · · ·  (t2k−2 , t2k−1 ) ⊂ (t0 , t1 ), we again time-order the operators and then multiply with evolution operators inserted for the closed intervals between them: the structure map for F is 



O1 ⊗ · · · ⊗ Ok → ei(t0 −t0 )H O1 ei(t2 −t1 )H · · · Ok ei(t1 −t2k−1 )H . Note that these structure maps reduce to those for an arbitrary associative algebra if we set H = 0 and hence have the identity map as the evolution operator. (In general, a one-parameter semigroup of algebra automorphisms can be used like this to “twist” the prefactorization algebra associated to an associative algebra.) As further examples, we have 

• The inclusion [0, t) ⊂ [0, t ) has structure map v0 | → v0 | ei(t −t)H .  • The inclusion (t, T] ⊂ (t , T] has structure map | v1 → ei(t−t )H | v1 . • The inclusion [0, t0 )  (t1 , t2 ) ⊂ [0, t ) has structure map v0 | ⊗ O →  v0 | ei(t1 −t0 )H Oei(t −t2 )H . All these choices ensure that we are free to choose what happens during open intervals but that the system evolves according to H during their closed complements. In this way, the prefactorization algebra encodes the basic abstract structure of quantum mechanics. The open interior (0, T) encodes the algebra of observables, and the boundaries encode the state spaces. Attentive readers might notice that we have not discussed, e.g., the structure map associated to (t0 , t1 ) ⊂ [0, t ). Here we need to use the distinguished elements in F((t0 , t1 )) = A and F([0, t )) = V arising from the inclusion of the

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empty set into these opens. If we fix vin | as the “idealized” initial state at time  0, then we set vin | eit H to be the distinguished element in F([0, t )) = V. The distinguished element of F((t0 , t1 )) is the evolution operator ei(t1 −t0 )H . Thus, the structure map for (t0 , t1 ) ⊂ [0, t ) is naturally 

O → vin | eit0 H Oei(t −t1 )H . We now describe the dual situation for intervals containing the other endpoint. Here we specify an “idealized” final state | vout so that the distinguished element of F((t, T]) is ei(T−t)H | vout . We do not need these initial or final states to recover the quantum mechanical formalism from the prefactorization algebra, so it is interesting that the prefactorization perspective pushes toward fixing these boundary states in the form of the distinguished elements. Remark: Although we have explained here how to start with the standard ingredients of quantum mechanics and encode them as a prefactorization algebra, one can also turn the situation around and motivate (or interpret), via the factorization perspective, aspects of the quantum mechanical formalism. For instance, time-reversal amounts to reflection across a point in R. Requiring a locally constant prefactorization algebra on R to be equivariant under timereversal corresponds to equipping the corresponding associative algebra A with an involutive algebra antiautomorphism. In other words, the prefactorization algebra corresponds to a ∗-algebra A. Likewise, suppose we want a prefactorization algebra on a closed interval [0, T] such that it corresponds to A on the open interior. The right end point corresponds to a left A-module V, viewed as the “outgoing” states. It is natural to want the “incoming” states – given by some right A-module V – to be “of the same type” as V (e.g., abstractly isomorphic) and to want the global sections to be C. This forces us to have a map V ⊗ A ⊗ V → C. This map induces a pairing between V and V, which provides a pre-Hilbert structure. ♦ Remark: Our construction above captures much of the standard formalism of quantum mechanics, but there are a few loose ends to address. First, in standard quantum mechanics, a state is not a vector in V but a line. Above, however, we fixed vectors vin | and | vout , so there seems to be a discrepancy. The observation that rescues us is a natural one, from the mathematical viewpoint. Consider scaling the initial and final states by elements of C× . This defines a new factorization algebra, but it is isomorphic

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to what we described earlier, and the expected value “ v0 |Ov1 ” of an operator depends linearly in the rescaling of the input and output vectors. More precisely, there is a natural equivalence relation we can place on the factorization algebras described earlier that corresponds to the usual notion of state in quantum mechanics. In other words, we could make a groupoid of factorization algebras where the underlying vector spaces and structure maps are all the same, but the distinguished elements are allowed to change. Another issue that might bother readers is that our formalism matches nicely only with experiments that resemble scattering experiments. It does not seem well suited to descriptions of systems such as bound states (e.g., an atom sitting quietly, minding its own business). For such systems, we might consider running over the whole space of states, which is described as a groupoid in the previous paragraph. Alternatively, we might drop the endpoints and simply work with the factorization algebra on the open interval, which focuses on the algebra of operators. ♦

3.4 A Construction of the Universal Enveloping Algebra Let g be a Lie algebra. In this section we describe a procedure that produces the universal enveloping algebra Ug as a prefactorization algebra on R. This construction is useful both because it is a model for a more general result about En algebras (see Section 6.4 in Chapter 6) and because it appears in several of our examples (such as the Kac–Moody factorization algebra). For a mathematician, it may be useful to see techniques similar to those we use in Section 4.2 in Chapter 4 shorn of any connection with physics, so that the underlying process is clearer. Let gR denote the cosheaf on R that assigns ( ∗c (U) ⊗ g, ddR ) to each open U, with ddR the exterior derivative. This is a cosheaf of cochain complexes, but it is only a precosheaf of dg Lie algebras. Note that the cosheaf axiom involves the use of coproducts, and the coproduct in the category of dg Lie algebras is not given by direct sum of the underlying cochain complexes. In fact, gR is a prefactorization algebra taking value in dg Lie algebras equipped with direct sum ⊕ as the symmetric monoidal product (here direct sum means the sum of the underlying cochain complexes, which inherits a natural Lie bracket). Let C∗ h denote the Chevalley–Eilenberg complex for Lie algebra homology, written as a cochain complex. In other words, C∗ h is the graded vector space Sym(h[1]) with a differential determined by the bracket of h. See Section A.4 in Appendix A for the definition and further discussion of this construction.

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Our main result shows how to construct the universal enveloping algebra Ug using C∗ (gR ). Proposition 3.4.1 Let H denote the cohomology prefactorization algebra of C∗ (gR ). That is, we take the cohomology of every open and every structure map, so H(U) = H ∗ (C∗ (gR (U))) for any open U. Then H is locally constant, and the corresponding associative algebra is isomorphic to Ug, the universal enveloping algebra of g. Remark: Recall Lemma 3.2.2, which says that every locally constant prefactorization algebra on R corresponds to an associative algebra. This proposition above provides a homological mechanism for recovering the universal enveloping algebra of a Lie algebra, but readers have probably noticed that we could apply the same construction with R replaced by any smooth manifold M. In Chapter 6, we will explain how to understand what this general procedure means. ♦ Proof Local constancy of H is immediate from the fact that, if I ⊂ J is the inclusion of one interval into another, the map of dg Lie algebras ∗c (I) ⊗ g → ∗c (J) ⊗ g is a quasi-isomorphism. We let Ag be the associative algebra constructed from H by Lemma 3.2.2. The underlying vector space of Ag is the space H(I) for any interval I. To be concrete, we will use the interval R, so that we identify Ag = H(R) = H ∗ (C∗ ( ∗c (R) ⊗ g)). We now identify that vector space. The dg Lie algebra ∗c (R) ⊗ g is concentrated in degrees 0 and 1 and maps quasi-isomorphically to its cohomology Hc∗ (R) ⊗ g = g[−1], which is concentrated in degree 1 by the Poincar´e lemma. This cohomology is an Abelian Lie algebra because the cup product on Hc∗ (I) is zero. It follows that C∗ ( ∗c (R)⊗g) is quasi-isomorphic to the Chevalley–Eilenberg chains of the Abelian Lie algebra g[−1], which is simply Sym g. Thus, as a vector space, Ag is isomorphic to the symmetric algebra Sym g. There is a map  : g → Ag that sends an element X ∈ g to ε ⊗X where ε ∈ Hc1 (I) is a basis element for the compactly supported cohomology of the interval I; we require the integral of ε

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to be 1. We will show that  is a map of Lie algebras, where Ag is given the Lie bracket coming from the associative structure. This result immediately implies the theorem, as we then have an map of associative algebra  : Ug → Ag that is an isomorphism of vector spaces. Let us check explicitly that  is a map of Lie algebras. Let δ > 0 be small ∞ number, and let f0 ∈ Cc (−δ, δ) be a compactly supported smooth function with f0 dx = 1. Let ft (x) = f0 (x − t). Note that ft is supported on the interval (t − δ, t + δ). If X ∈ g, a cochain representative for (X) ∈ Ag is provided by f0 dx ⊗ X ∈ 1c ((−δ, δ)) ⊗ g. Indeed, because every ft dx is cohomologous to f0 dx in 1c (R), the element ft dx ⊗ X is a cochain representative of (X) for any t. Given elements α, β ∈ Ag , the product α · β is defined as follows. (i) We choose intervals I, J with I < J. (ii) We regard α as an element of H(I) and β as an element of H(J) using the inverses to the isomorphisms H(I) → H(R) and H(J) → H(R) coming from the inclusions of I and J into R. (iii) The product α · β is defined by taking the image of α ⊗ β under the factorization structure map H(I) ⊗ H(J) → H(R) = Ag . Let us see how this works with our representatives. The cohomology class [ft dx ⊗ X] ∈ H(t − δ, t + δ) becomes (X) under the natural map from H(t − δ, t + δ) to H(R). If we take δ to be sufficiently small, the intervals (−δ, δ) and (1 − δ, 1 + δ) are disjoint. It follows that the product (X)(Y) is represented by the cocycle (f0 dx ⊗ X)(f1 dx ⊗ Y) ∈ Sym2 ( 1c (R) ⊗ g) ⊂ C∗ ( ∗c (R) ⊗ g). Similarly, the commutator [(X), (Y)] is represented by the expression (f0 dx ⊗ X)(f1 dx ⊗ Y) − (f0 dx ⊗ Y)(f−1 dx ⊗ X). It suffices to show that this cocycle in C∗ ( ∗c (R) ⊗ g) is cohomologous to ([X, Y]). Note that the 1-form f1 dx − f−1 dx has integral 0. It follows that there exists a compactly supported function h ∈ Cc∞ (R) with ddR h = f−1 dx − f1 dx. We can assume that h takes value 1 in the interval (−δ, δ).

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We now calculate the differential of the element (f0 dx ⊗ X)(h ⊗ Y) ∈ C∗ ( ∗c (R) ⊗ g). We have d ((f0 dx ⊗ X)(h ⊗ Y)) = (f0 dx ⊗ X)(dh ⊗ Y) + f0 h dx ⊗ [X, Y] = (f0 dx ⊗ X)((f−1 − f1 )dx ⊗ Y) + f0 h dx ⊗ [X, Y]. Since h takes value 1 on the interval (−δ, δ), we know f0 h = f0 . This equation tells us that a representative for [(X), (Y)] is cohomologous to ([X, Y]).

3.5 Some Functional Analysis Nearly all of the examples of factorization algebras that we will consider in this book will assign to an open subset, a cochain complex built from vector spaces of analytical provenance: for example, the vector space of smooth sections of a vector bundle or the vector space of distributions on a manifold. Such vector spaces are best viewed as being equipped with an extra structure, such as a topology, reflecting their analytical origin. In this section we briefly sketch a flexible multicategory of vector spaces equipped with an extra “analytic” structure. More details are contained in Appendix B.

3.5.1 Differentiable Vector Spaces The most common way to encode the analytic structure on a vector space such as the space of smooth functions on a manifold is to endow it with a topology. Homological algebra with topological vector spaces is not easy, however. (For instance, topological vector spaces do not form an Abelian category.) To get around this issue, we will work with differentiable vector spaces. Let us first define the slightly weaker notion of a C∞ -module. Definition 3.5.1 Let Mfld be the site of smooth manifolds, i.e., the category of smooth manifolds and smooth maps between them, where a cover is a surjective local diffeomorphism. Let C∞ denote the sheaf of rings on Mfld that assigns to any manifold M the commutative algebra C∞ (M). A C∞ -module is a module sheaf over C∞ on Mfld. In other words, to each manifold M, a C∞ -module F assigns a module F(M) over the algebra C∞ (M), and for any map of manifolds f : M → N, the

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pullback map f ∗ : F(N) → F(M) is a map of C∞ (N)-modules. For example, if V is any topological vector space, then there is a natural notion of smooth map from any manifold M to V (see, e.g., Kriegl and Michor (1997)). The space C∞ (M, V) of such smooth maps is a module over the algebra C∞ (M). Since smoothness is a local condition on M, sending M to C∞ (M, V) gives a sheaf of C∞ -modules on the site Mfld. As an example of this construction, let us consider the case when V is the space of smooth functions on a manifold N, equipped with its usual Fr´echet topology. One can show that for each manifold M, the space C∞ (M, C∞ (N)) is naturally isomorphic to the space C∞ (M×N) of smooth functions on M×N. As we will see shortly, we lose very little information when we view a topological vector space as a C∞ -module. Sheaves of C∞ -modules on Mfld that arise from topological vector spaces are endowed with an extra structure: we can always differentiate smooth maps from a manifold M to a topological vector space V. Differentiation can be viewed as an action of the vector fields on M on the vector space C∞ (M, V). Dually, it can be viewed as coming from a connection ∇ : C∞ (M, V) → 1 (M, V), where 1 (M, V) is defined to be the tensor product 1 (M, V) = 1 (M) ⊗C∞ (M) C∞ (M, V). This tensor product is just the algebraic one, which is well behaved because 1 (M) is a projective C∞ (M)-module of finite rank. This connection is flat, in that the curvature F(∇) = ∇ ◦ ∇ : C∞ (M, V) → 2 (M, V) is zero. Flatness ensures that the action of vector fields is actually a Lie algebra action, so that we get an action of all differential operators on M and not just vector fields. This flatness property suggests the following definition. Definition 3.5.2 Let 1 denote the C∞ -module that assigns 1 (M) to a manifold M. For F a C∞ -module, the C∞ -module of k-forms valued in F is the C∞ module k (F) that assigns to a manifold M, the tensor product k (M, F) = k (M) ⊗C∞ (M) F(M). A connection on a C∞ -module F is a map of sheaves on the site Mfld, ∇ : F → 1 (F),

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that satisfies the Leibniz rule on every manifold M. A connection is flat if it is flat on every manifold M. A differentiable vector space is a C∞ -module equipped with a flat connection. A map of differentiable vector spaces f : F → G is a map of C∞ -modules that intertwines with the flat connections. We denote the set of all such maps by DVS(F, G). Almost all of the differentiable vector spaces we will consider are concrete in nature. Indeed, most satisfy the formal definition of a being a concrete sheaf, which we now explain. Let Set denote the category of sets and let Set(S, T) denote the collection of functions from the set S to the set T. For any sheaf F on Mfld taking values in Set, there is a natural map F(M) → Set(M, F(∗)): each element of the set F(M) has an associated function from the underlying set M to the set F(∗), the value of the sheaf on a point. We say a sheaf F is concrete if this map F(M) → Set(M, F(∗)) is injective. Hence, for a concrete sheaf, one can think of any section on M as just a particular function from M to F(∗); a section is just a “smooth” function on M with values in the set F(∗). As an example of a concrete sheaf, consider the sheaf X associated to a smooth manifold X, where X(M) = C∞ (M, X). In this case, a section of the sheaf X really is just a function to X. This sheaf just identifies which set-theoretic functions are smooth. We often work with the differentiable vector space arising from a topological vector space V, which, just like the example X, simply records which set-theoretic maps are smooth. For this reason, we will normally think of a differentiable vector space F as being an ordinary vector space, given by its value on a point F(∗), together with extra structure. We will often refer to the sections F(M) (i.e., the value of the sheaf F on the manifold M) as the space of smooth maps to the value on a point. If V is a differentiable vector space, we often write C∞ (M, V) for the space of smooth maps from a manifold M. Abusively, we will also often call a map of differentiable vector spaces simply a smooth map.

3.5.2 Differentiable Cochain Complexes Although we typically work with differentiable vector spaces coming from topological vector spaces, the category DVS is much easier to use – for our purposes – than that of topological vector spaces. The key reason is that they are (essentially) just sheaves on a site, and homological algebra for such objects is very well developed, as we explain in Appendix C.

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Definition 3.5.3 A differentiable cochain complex is a cochain complex in the category of differentiable vector spaces. A cochain map f : V → W of differentiable cochain complexes is a quasiisomorphism if the map C∞ (M, V) → C∞ (M, W) is a quasi-isomorphism for all manifolds M. This condition is equivalent to asking that the map be a quasiisomorphism at the level of stalks. We use Ch(DVS) to denote the category of differentiable cochain complexes and cochain maps. It can be enriched over cochain complexes of vector spaces in the usual way.

3.5.3 Differentiable Prefactorization Algebras We have defined the notion of prefactorization algebra with values in any multicategory. To discuss factorization algebras valued in differentiable vector spaces, we need to define a multicategory structure on differentiable vector spaces. Let us first discuss the multicategory structure on the C∞ -modules. Definition 3.5.4 Let V1 , . . . , Vn , W be differentiable vector spaces. A smooth multilinear map  : V1 × · · · × Vn → W is a C∞ -multilinear map  of sheaves that satisfies the following Leibniz rule with respect to the connections on the Vi and W. For every manifold M, and for every vi ∈ Vi (M), we require that ∇(v1 , . . . , vn ) =

n 

(v1 , . . . , ∇vi , . . . , vn ) ∈ 1 (M, W).

i=1

We let DVS(V1 , . . . , Vn | W) denote this space of smooth multilinear maps. In more down-to-earth terms, such a  is a C∞ (M)-multilinear map (M) : V1 (M) × · · · × Vn (M) → W(M) for every manifold M, in a way compatible with the connections and with the maps Vi (M) → Vi (N) associated to a map f : N → M of manifolds. The category of differentiable cochain complexes acquires a multicategory structure from that on differentiable vector spaces, where the multimaps are smooth multilinear maps that are compatible with the differentials. (Here “compatible” means precisely the same thing as it does with ordinary vector spaces and cochain complexes. It means, in the case of unary maps, that we work with cochain maps and not arbitrary linear maps of graded vector spaces.)

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Definition 3.5.5 A differentiable prefactorization algebra is a prefactorization algebra valued in the multicategory of differentiable cochain complexes. In words, a differentiable prefactorization algebra F on a space X assigns a differentiable cochain complex F(U) to every open subset U ⊂ X, and a smooth multilinear cochain map F(U1 ) × · · · × F(Un ) → F(V), whenever U1 , . . . , Un are disjoint opens contained in V. Note the very different roles played here by the space X, on which the prefactorization algebra lives, and the site Mfld of manifolds, on which the differentiable cochain complexes live. The topology of the space X organizes the algebraic structure we are interested in. By contrast, the geometry of manifolds encoded in Mfld organizes the structure of the vector spaces (or cochain complexes) we work with. Said succinctly, as a substitute for a topology on these vector spaces, we use a sheaf structure over Mfld.

3.5.4 Relationship with Topological Vector Spaces As we have seen, every topological vector space gives rise to a differentiable vector space. There is a beautiful theory developed in Kriegl and Michor (1997) concerning the precise relationship between topological vector spaces and differentiable vector spaces. These results are discussed in much more detail in Appendix B; we briefly summarize them now. Let LCTVS denote the category of locally convex Hausdorff topological vector spaces, and continuous linear maps. Let BVS denote the category with the same objects, but whose morphisms are bounded linear maps. Every continuous linear map is bounded, but not conversely. These categories have natural enrichments to multicategories, where the multimaps are continuous (respectively, bounded) multilinear maps. The category BVS is equivalent to a full subcategory of LCTVS whose objects are called bornological vector spaces. Theorem 3.5.6 The functor dift : LCTVS → DVS restricts to a functor difβ : BVS → DVS, which embeds BVS as a full sub-multicategory of DVS. In other words: if V, W are topological vector spaces, and if dift (V), dift (W) denote the corresponding differentiable vector spaces, then the maps in DVS from dift (V) to dift (W) are the same as bounded linear maps from V to W. More generally, if V1 , . . . , Vn and W are topological vector spaces, the bounded multilinear maps BVS(V1 , . . . , Vn | W) are the same as smooth multilinear maps DVS(dift (V1 ), . . . , dift (Vn ) | dift (W)).

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This theorem tells us that we lose very little information if we think of a topological vector space as being a differentiable vector space. We just end up thinking about bounded maps rather than continuous maps. Remark: On occasion, we will exploit this relationship by using a concept from topological or bornological vector spaces without profferring a differentiable analog. For example, in a few places we talk about a dense inclusion and use it to show that a map out of the bigger space is determined by the subspace. In such cases, we are working only with bornological vector spaces and depend♦ ing upon the fact that difβ is a full and faithful functor. So far, however, we have not discussed how completeness of topological vector spaces appears in this context. We need a notion of completeness for a topological vector space that depends only on smooth maps to that vector space. The relevant concept was developed in Kriegl and Michor (1997). We will view the category BVS as being a full subcategory of DVS. Definition 3.5.7 A topological vector space V ∈ BVS is c∞ -complete, or convenient, if every smooth map c : R → V has an antiderivative. We denote the category of convenient vector spaces and bounded linear maps by CVS. This completeness condition is a little weaker than the one normally studied for topological vector spaces. That is, every complete topological vector space is c∞ -complete. Proposition 3.5.8 The full subcategory difc : CVS ⊂ DVS is closed under the formation of limits, countable coproducts, and sequential colimits of closed embeddings. We give CVS the multicategory structure inherited from BVS. Since BVS is a full sub-multicategory of DVS, so is CVS. Theorem 3.5.9 The multicategory structure on CVS is represented by a symβ . metric monoidal structure ⊗ This symmetric monoidal structure is called the completed bornological tensor product. If E, F ∈ CVS, this completed bornological tensor product β F. The statement that it represents the multicategory strucis written as E⊗ ture means that smooth (equivalently, bounded) bilinear maps f : E1 ×E2 → F β E2 → F, for objects E1 , E2 , F are the same as bounded linear maps f  : E1 ⊗ of CVS.

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When it should cause no confusion, we may use the symbol ⊗ instead of β for this tensor product. ⊗

3.5.5 Examples from Differential Geometry Let us now give some examples of differentiable vector spaces. These examples will include the basic building blocks for most of the factorization algebras we will consider. Let E be a vector bundle on a manifold X. We let E (X) denote the vector space of smooth sections of E on X, and we let Ec (X) denote the vector space of compactly supported sections of E on X. Let us give these vector spaces the structure of differentiable vector spaces, as follows. If M is a manifold, we say a smooth map from M to E is a smooth section of the bundle πX∗ E on M × X. We denote this set of smooth maps C∞ (M, E (X)). Sending M to C∞ (M, E (X)) defines a sheaf of C∞ -modules on the site of smooth manifolds with a flat connection, and so a differentiable vector space. Similarly, we say a smooth map from M to Ec (X) is a smooth section of the bundle πX∗ E on M × X, whose support maps properly to M. Let us denote this set by C∞ (M, Ec (X)); this defines, again, a sheaf of C∞ -modules on the site of smooth manifolds with a flat connection. Theorem 3.5.10 With this differentiable structure, the spaces E (X) and Ec (X) are in the full subcategory CVS of convenient vector spaces. Further, this differentiable structure is the same as the one that arises from the natural topologies on E (X) and Ec (X). The proof (like the proofs of all results in this section) are contained in Appendix B, and based heavily on the book Kriegl and Michor (1997). Let E be a vector bundle on a manifold X. Throughout this book, we will often use the notation E (X) to denote the distributional sections on X, defined by E (X) = E (X) ⊗C∞ (X) D(X), where D(X) is the space of distributions on X. Similarly, let E c (X) denote the compactly supported distributional sections of E on X. There are natural inclusions Ec (X) → E c (X) → E (X), Ec (X) → E (X) → E (X), by viewing smooth functions as distributions.

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Let E! denote the vector bundle E∨ ⊗ DensX , which possesses a natural vector bundle pairing ev : E ⊗ E! → DensX . In other words, a smooth section of E and a smooth section of E! can be paired to produce a smooth density on X. If this smooth density has compact support, it can certainly be integrated to produce a real number. With a little more work, one can show that E c (X) is the continuous dual to E ! (X), the space of smooth sections of E! on X. Likewise, ! one can show that Ec (X) is the continuous dual to E (X), the distributional sections of E! . These topological vector spaces E (X) and E c (X) thus obtain natural differentiable structures via the functor dift . We have the following description of the differentiable structure. Theorem 3.5.11 For any manifold M, a smooth map from M to E (X) is the same as a smooth linear map Ec! (X) → C∞ (M). Similarly, a smooth map from M to E c (X) is the same as a smooth linear map E ! (X) → C∞ (M). This is a consequence of Lemma B.5.2.2. See also Section B.2.1 in Appendix B for more discussion.

3.5.6 Multilinear Maps and Enriched Spaces of Maps The category of differentiable vector spaces has a natural tensor product. In other words, it is a symmetric monoidal category. The tensor product in DVS is very simple: if V, W are differentiable vector spaces, then they are C∞ modules (by forgetting the flat connections) so V ⊗C∞ W is another C∞ module, but it inherits a natural flat connection ∇V ⊗ IdW + IdV ⊗∇W , so we obtain a new differentiable vector space. One must be careful with this tensor product, however. From the point of view of analysis, this tensor product is not very meaningful: it is somewhat similar to an uncompleted tensor product of topological vector spaces. For example, if M and N are manifolds, then C∞ (M) and C∞ (N) naturally have the structure of differentiable vector spaces. It is not true that C∞ (M) ⊗C∞ C∞ (N) = C∞ (M × N).

(†)

This issue means that our examples will not assign to a disjoint union of opens the tensor product of values on the components. The multicategory structure on DVS that we use coincides with this symmetric monoidal structure. The multicategory structure is better behaved than the

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symmetric monoidal structure: when restricted to the full subcategory CVS the multicategory becomes the one associated to the symmetric monoidal structure on CVS, which has good analytical properties (and in particular satisfies the equality in (†)). Similarly, there is an internal hom in the category of C∞ -modules, and this sheaf hom likewise inherits a natural flat connection. We denote this sheaf hom by HomDVS (V, W). This sheaf hom is not as well behaved as one would hope, however, and does not capture the concept of “smooth families of maps.” In particular, it is not true that the value of the sheaf HomDVS (V, W) on a point is the vector space DVS(V, W) of smooth maps from V to W. For any reasonable definition of the notion of smooth family of maps parametrized by a manifold, a smooth family of maps parametrized by a point should be simply a map, and the self-enrichment given by HomDVS (V, W) does not satisfy this. There is, however, another way to enrich the category DVS over itself that better captures the notion of smooth family of maps. (For a careful treatment, see Section B.6 in Appendix B.) Before we define this enrichment, we need the following definition. Definition 3.5.12 For V a differentiable vector space and M a manifold, let C∞ (M, V) denote the differentiable vector space whose value on a manifold N is C∞ (N × M, V). The flat connection map ∇N,C∞ (M,V) : C∞ (N, C∞ (M, V)) → 1 (N, C∞ (M, V)) is the composition of the flat connection ∇N×M : C∞ (M × N, V) → 1 (M × N, V) = 1 (M, C∞ (N, V)) ⊕ 1 (N, C∞ (M, V)) with the projection onto 1 (N, C∞ (M, V)). Here the direct sum decomposition of 1-forms is a consequence of the fact ∗ ∗ T ∗ ⊕ π ∗ T ∗ , where π : M × N → M splits as the direct sum πM that TM×N M M N N and πN : M × N → N denote the projection maps. This definition makes the category of differentiable vector spaces into a category cotensored over the category of smooth manifolds. Note that C∞ (M) defines a ring object in the category DVS, and for any differentiable vector space V, the mapping space C∞ (M, V) is a C∞ (M)-module. This construction generalizes in a natural way. If E is a vector bundle on a manifold M and V is a differentiable vector space, then we can define C∞ (M, E ⊗ V) := E (M) ⊗C∞ (M) C∞ (M, V)

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where on the right-hand side we are taking tensor products in the category DVS over the differentiable ring C∞ (M). Although, in general, tensor products in DVS are not analytically well behaved, in this case there are no problems because E (M) is a projective C∞ (M) module of finite rank. We will denote C∞ (M, T ∗ M ⊗ E) by 1 (M, E). Definition 3.5.13 Let V1 , . . . , Vn , W be differentiable vector spaces. Given a manifold M, a smooth family of multilinear maps V1 × · · · × Vn → W parametrized by M is an element of DVS(V1 , . . . , Vn | C∞ (M, W)) where we regard C∞ (M, W) as being a differentiable vector space using the definition given earlier. In Section B.6 of Appendix B we shown that there is a differentiable vector space HomDVS (V1 , . . . , Vn | W) defined by saying that its value on a manifold M is DVS(V1 , . . . , Vn | C∞ (M, W)). The flat connection comes from the natural map ∇ C∞ (M,W) : C∞ (M, W) → 1 (M, W), given by applying the projection in the M-direction to the connection ∇M×N,W (compare to the connection on C∞ (M, W)). This definition makes the category DVS into a multicategory enriched in itself, in such a way that if we evaluate the differentiable Hom-space on a point, we recover the original vector space of smooth maps. In this text, whenever we consider a smooth family of maps between differentiable vector spaces, we are always referring to this self-enrichment.

3.5.7 Algebras of Observables The prefactorization algebras we will use for most of the book are built as algebras of functions on the convenient vector spaces E (X), which for us will mean symmetric algebras on their dual spaces. Recall that E (X) and E c (X) are both convenient vector spaces, and that, in the full subcategory CVS ⊂ DVS of convenient vector spaces, the multicategory is represented by a symmetric π denotes the completed projecβ . Recall as well that ⊗ monoidal structure ⊗ tive tensor product in LCTVS. Proposition 3.5.14 Let E be a vector bundle on a manifold X, and let F be a vector bundle on a manifold Y. Let E (X) denote the convenient vector space of

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smooth sections of E on X, and let F (Y) denote the convenient vector space of smooth sections of F on Y. Then π F (Y), β F (Y) ∼ E (X) ⊗ = (X × Y, E  F) ∼ = E (X) ⊗ where E  F denotes the external tensor product of vector bundles and  denotes smooth sections. Remark: An alternative approach to the one we’ve taken is to use the category of nuclear topological vector spaces, with the completed projective tensor product, instead of the category of convenient (or differentiable) vector spaces. Using nuclear spaces raises a number of technical issues, but one immediate π C∞ (Y) = C∞ (X × Y), issue is the following: although it is true that C∞ (X)⊗ it seems not to be true that the same statement holds if we use compactly supported smooth functions. The problem stems from the fact that the projective tensor product does not commute with colimits, whereas the bornological tensor product does. ♦ We can define symmetric powers of convenient vector spaces using the symmetric monoidal structure we have described. If, as before, E is a vector bundle on X and U is an open subset of X, this proposition allows us to identify Symn (Ec (U)) = Cc∞ (U n , En )Sn . The symmetric algebra Sym Ec (U) is defined as usual to be the direct sum of the symmetric powers. It is an algebra in the symmetric monoidal category of convenient vector spaces. A related construction is the algebra of functions on a differentiable vector space. If V is a differentiable vector space, we can define, as we have seen, the space of linear functionals on V to be the space of maps HomDVS (V, R). Because the category DVS is self-enriched, this is again a differentiable vector space. In a similar way, we can define the space of polynomial functions on V homogeneous of degree n to be the space Pn (V) = HomDVS (V, . . . , V | C∞ )Sn .    n times

In other words, we take smooth multilinar maps from n copies of V to R, and then take the Sn coinvariants. The self-enrichment of DVS gives this the structure of differentiable vector space. Concretely, a smoooth map from a manifold M to Pn (V) is C∞ (M, Pn (V)) = HomDVS (V, . . . , V | C∞ (M))Sn .    n times

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One can then define the algebra of functions on V by  O(V) = Pn (V). n

(In this formula, we take the product rather than the direct sum, so that O(V) should be thought of as a space of formal power series on V. One can, of course, also consider the version using the sum.) The space O(V) is a commutative algebra object of the category DVS in a natural way. This construction is a very general one, of course: one can define the algebra of functions on any object in any multicategory in the same way. An important example is the following. Lemma 3.5.15 Let E be a vector bundle on a manifold X. Then, Pn (E(X)) is isomorphic as a differentiable vector space to the Sn covariants of the space of compactly supported distributional sections of E! on X n . Proof We know that the multicategory structure on the full subcategory CVS ⊂ DVS is represented by a symmetric monoidal category, and that in this symmetric monoidal category, 

E (X)⊗β n = (X n , En ). It follows from this that, for any manifold M, we can identify C∞ (M, Pn (E (X))) with the Sn covariants of the space of smooth linear maps (X n , En ) → C∞ (M). In fact, because the spaces in this equation are bornological (see Appendix B), such smooth linear maps are the same as continuous linear maps. We have seen that this space of smooth linear maps is – with its differentiable structure – the same as the space Dc (X n , (E! )n ) of compactly supported distributional sections of the bundle (E! )n . Note that O(E (U)) is naturally the same as HomDVS (Sym E (U), R), i.e., it is the dual of the symmetric algebra of E (U).

3.6 The Factorization Envelope of a Sheaf of Lie Algebras In this section, we will introduce an important class of examples of factorization algebras. We will show how to construct, for every fine sheaf of Lie algebras L on a manifold M, a factorization algebra that we call the factorization envelope. This construction is our version of the chiral envelope introduced in

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Beilinson and Drinfeld (2004). The construction can also be viewed as a natural generalization of the universal enveloping algebra of a Lie algebra. Indeed, we have shown in Section 3.4 that the factorization envelope of the constant sheaf of Lie algebras g on R is the universal enveloping algebra of g, viewed as a factorization algebra on R. The factorization envelope plays an important role in our story. (i) The factorization algebra of observables for a free field theory is an example of a factorization envelope. (ii) In Section 5.5 in Chapter 5, we will show, following Beilinson and Drinfeld, that the Kac–Moody vertex algebra arises as a (twisted) factorization envelope. The most important appearance of factorization envelopes appears in our treatment of Noether’s theorem at the quantum level, which is covered in Volume 2. We show there that if a sheaf of Lie algebras L acts on a quantum field theory on a manifold M, then there is a morphism from a twisted factorization envelope of L to the quantum observables of the field theory. This construction is very useful. For example, for a chiral conformal field theory, it allows one to construct a map of factorization algebras from a Virasoro factorization algebra to the quantum observables. This map induces a map of vertex algebras via our construction in Section 5.3 in Chapter 5.

3.6.1 The Key Idea Thus, let M be a manifold. Let L be a fine sheaf of dg Lie algebras on M. Let Lc denote the cosheaf of compactly supported sections of L. (We restrict to fine sheaves so that taking compactly supported sections is a straightforward operation.) Remark: Note that, although Lc is a cosheaf of cochain complexes, and a precosheaf of dg Lie algebras, it is not a cosheaf of dg Lie algebras. This is because colimits of dg Lie algebras are not the same as colimits of cochain complexes. ♦ We can view Lc as a prefactorization algebra valued in the category of dg Lie algebras with symmetric monoidal structure given by direct sum. Indeed, if {Ui } is a finite collection of disjoint opens in M contained in the open V, there is a natural map of dg Lie algebras  Lc (Ui ) = Lc (i Ui ) → Lc (V) i

giving the factorization product.

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Taking Chevalley–Eilenberg chains is a symmetric monoidal functor from dg Lie algebras, equipped with the direct sum monoidal structure, to cochain complexes. Definition 3.6.1 If L is a sheaf of dg Lie algebras on M, the factorization envelope UL is the prefactorization algebra obtained by applying the Chevalley–Eilenberg chain functor to Lc , viewed as a prefactorization algebra valued in dg Lie algebras. Concretely, UL assigns to an open subset V ⊂ M the cochain complex UL(V) = C∗ (Lc (V)), where C∗ is the Chevalley–Eilenberg chain complex. The factorization structure maps are defined as follows: given a finite collection of disjoint opens {Ui } in V, we have     ∼ C∗ C∗ Lc (Ui ) = Lc (Ui ) → C∗ Lc (V). i

i

This construction is parallel to the example in Section 3.1.1. We will see later (see Theorem 6.5.3 in Chapter 6) that this prefactorization algebra is a factorization algebra.

3.6.2 Local Lie Algebras In practice, we will need an elaboration of this construction that involves a small amount of analysis. Definition 3.6.2 Let M be a manifold. A local dg Lie algebra on M consists of the following data: (i) A graded vector bundle L on M, whose sheaf of smooth sections will be denoted L. (ii) A differential operator d : L → L, of cohomological degree 1 and square 0. (iii) An alternating bi-differential operator [−, −] : L⊗2 → L that endows L with the structure of a sheaf of dg Lie algebras. Remark: This definition will play an important role in our approach to interacting classical field theories, developed in Volume 2. ♦

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If U ⊂ M, then L(U) is a topological vector space, because it is the space of smooth sections of a graded vector bundle on U. We would like to form, as above, the Chevalley–Eilenberg chain complex C∗ (Lc (U)). The underlying vector space of C∗ (Lc (U)) is the (graded) symmetric algebra on Lc (U)[1]. We need to take account of the topological structure on Lc (U) when we take the tensor powers of Lc (U). We explained how to do this in Section 3.5: we define (Lc (U))⊗n to be the tensor power defined using the completed projective tensor product on the topological vector space Lc (U). Concretely, if Ln denotes the vector bundle on M n obtained as the external tensor product, then (Lc (U))⊗n = c (U n , Ln ) is the space of compactly supported smooth sections of Ln on U n . Symmetric (or exterior) powers of Lc (U) are defined by taking coinvariants of Lc (U)⊗n with respect to the action of the symmetric group Sn . The completed symmetric algebra on Lc (U)[−1] that is the underlying graded vector space of C∗ (Lc (U)) is defined using these completed symmetric powers. The Chevalley–Eilenberg differential is continuous, and therefore defines a differential on the completed symmetric algebra of Lc (U)[−1], giving us the cochain complex C∗ (Lc (U)). Example: Let g be a Lie algebra. Consider the sheaf ∗R ⊗ g of dg Lie algebras on R, which assigns to the open U the dg Lie algebra ∗ (U) ⊗ g. Then, as we saw in detail in Section 3.4, the factorization envelope of this sheaf of Lie algebras encodes the universal enveloping algebra Ug of g. Indeed, factorization algebras on R with an additional “locally constant” property give rise to associative algebras, and the associative algebra associated to U( ∗R ⊗ g) recovers the ordinary universal enveloping algebra. In the same way, for any Lie algebra g we can construct a factorization algebra on Rn as the factorization envelope of ∗Rn ⊗ g. The resulting factorization algebra is locally constant: it has the property that the inclusion map from one disc to another is a quasi-isomorphism. A theorem of Lurie (2016) tells us that locally constant factorization algebras on Rn are the same as En algebras. The En algebra we have constructed is the En -enveloping algebra of g. (See the discussion in Section 6.4 in Chapter 6 for more about locally constant factor♦ ization algebras, En algebras, and the En enveloping functor.)

3.6.3 Shifted Central Extensions and the Twisted Envelope Many interesting factorization algebras – such as the Kac–Moody factorization algebra, and the factorization algebra associated to a free field theory – can be

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constructed from a variant of the factorization envelope construction, which we call the twisted factorization envelope. Definition 3.6.3 Let L be a sheaf of dg Lie algebras on M. A k-shifted central c fitting into an exact extension of Lc is a precosheaf of dg Lie algebras L sequence c → Lc → 0 0 → C[k] → L of precosheaves, where C[k] is the constant presheaf that assigns the onedimensional vector space C[k] in degree −k to every open. If L is a local Lie algebra, we require in addition that locally there is a splitting c (U) = C[k] ⊕ Lc (U) L such that the differential and bracket maps from Lc (U) → C[k] and Lc (U)⊗2 → C[k] are continuous. Definition 3.6.4 In this situation, the twisted factorization envelope is the prefc (U)). (In the case actorization algebra U L that sends an open set U to C∗ (L that L is a local dg Lie algebra, we use the completed tensor product as above.) c (U)) is a module over chains on the Abelian Lie The chain complex C∗ (L algebra C[k] for every k. Thus, we will view the twisted factorization envelope as a prefactorization algebra in modules for C[c] where c has degree −k − 1. Under the assumption that L is a homotopy sheaf, Theorem 6.6.1 shows that the twisted factorization envelope U Ł is a factorization algebra over the base ring C[c]. Of particular interest is the case when k = −1, so that the central parameter c is of degree 0. Let us now introduce some important examples of this construction. Example: Let g be a Lie algebra and consider the local Lie algebra ∗R ⊗ g on the real line R, which we will denote gR . Given a skew-symmetric, invariant bilinear form ω on g, there is a natural shifted extension of gR c where  α ∧ β ω(X, Y) c, [α ⊗ X, β ⊗ Y]ω = α ∧ β ⊗ [X, Y] + R

where we use c to denote the generator in degree 1 of the central extension. Let Uω gR denote the twisted factorization envelope for this central extension. By mimicking the proof of Proposition 3.4.1, one can see that the cohomology of this twisted factorization envelope recovers the enveloping algebra U g of the central extension of g given by ω. ♦

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Example: Let g be a simple Lie algebra, and let −, − g denote a symmetric invariant pairing on g. We define the Kac–Moody factorization algebra as follows. Let  be a Riemann surface, and consider the local Lie algebra 0,∗ ⊗ g on , which sends an open subset U to the dg Lie algebra 0,∗ (U) ⊗ g. The ¯ so we are describing the Dolbeault analog of the de Rham differential here is ∂, construction in Section 3.4. There is a −1-shifted central extension of 0,∗ c ⊗ g defined by the cocycle  ω(α, β) = U

α, ∂β g

0,∗ → 1,∗ is the holomorphic de Rham where α, β ∈ 0,∗ c (U) ⊗ g and ∂ : operator. Note that this is a −1-shifted cocycle because ω(α, β) is zero unless deg(α) + deg(β) = 1. The twisted factorization envelope Uω ( 0,∗ ⊗ g) is the Kac–Moody factorization algebra. Locally, it recovers the Kac–Moody vertex algebra, just as the real one-dimensional case recovers the enveloping algebra. (We show this in Section 5.5 in Chapter 5.) ♦

Example: In this example we will define a higher-dimensional analog of the Kac–Moody vertex algebra. Let X be a complex manifold of dimension n. Let ∗ (X) denote the de Rham complex. Let φ ∈ n−1,n−1 (X) be a closed form. Then, given any Lie algebra g equipped with an invariant pairing −, − g ,  of L = 0,∗ we can construct a −1-shifted central extension L c ⊗ g, defined as above by the cocycle  ω : α ⊗ β → X

α, ∂β g ∧ φ.

It is easy to verify that ω is a cocycle. (The case of the Kac–Moody extension is when n = 1 and φ is a constant.) The cohomology class of this cocycle is unchanged if we change φ to φ + ∂ψ, where ψ ∈ n−1,n−2 (X) satisfies ∂ψ = 0.  = Uω L is closely related to the Kac– The twisted factorization envelope UL Moody algebra. For instance, if X =  × Pn−1 where  is a Riemann surface, and the form φ is a volume form on Pn−1 , then the pushforward of this factorization algebra to  is quasi-isomorphic to the Kac–Moody factorization algebra described earlier. (Let p : X →  denote the projection map. The

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pushforward is defined by (p∗ F)(U) = F(p−1 (U)) for each U ⊂ .) There is an important special case of this construction. Let X be a complex surface (i.e., dimC (X) = 2) and the form φ is the curvature of a connection on the canonical bundle of X (i.e., φ represents c1 (X)). As we will show when we discuss Noether’s theorem at the quantum level, if we have a field theory with an action of a local dg Lie algebra L then a twisted factorization envelopes of Ł will map to the factorization algebra of observables of the theory. One can show that the local dg Lie algebra 0,∗ X ⊗ g acts on a twisted N = 1 gauge theory with matter, and that the twisted factorization envelope – with central extension determined by c1 (X) – maps to observables of this theory (following Johansen (1995)). ♦

3.7 Equivariant Prefactorization Algebras Let M be a topological space with an action of a group G by homeomorphisms. For g ∈ G and U ⊂ M, we use gU to denote the subset {gx | x ∈ U} ⊂ M. We will formulate what it means to have a G-equivariant prefactorization algebra on M. When M is a manifold and G is a Lie group acting smoothly on M, we will formulate the notion of a smoothly G-equivariant prefactorization algebra on M.

3.7.1 Discrete Equivariance We begin with the case where G is viewed simply as a group (i.e., we do not require any compatibility between the action and a possible smooth structure on G). We give a concrete definition. Definition 3.7.1 Let F be a prefactorization algebra on M. We say F is Gequivariant if for each g ∈ G and each open subset U ⊂ M we are given isomorphisms ∼ =

→ F(gU) σg : F(U) − satisfying the following conditions. (i) For all g, h ∈ G and all opens U, σg ◦ σh = σgh : F(U) → F(ghU).

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(ii) Every σg respects the factorization product. That is, for any finite tuple U1 , . . . , Uk of disjoint opens contained in an open V, the diagram F(U1 ) ⊗ · · · ⊗ F(Un )

/ F(gU1 ) ⊗ · · · ⊗ F(gUn )

 F(V)

 / F(gV)

commutes. There is a more categorical way to phrase this definition. As G acts continuously, each element g ∈ G provides an endofunctor of the category OpensM of opens in M. Indeed, g provides an endofunctor of DisjM and hence, via precomposition, provides an endofunctor on the category of algebras over DisjM : it sends a prefactorization algebra A to a prefactorization algebra gA, where gA(U) = A(gU). An equivariant prefactorization algebra is then a collection of isomorphisms σg : A → gA of prefactorization algebras such that (hσg ) ◦ σh = σgh .

3.7.2 A Useful Reinterpretation We want to be able to talk about a prefactorization algebra F that is equivariant with respect to the smooth action on a manifold M of a Lie group G. In particular, we need to formulate what it means to give a “smooth” action of G on a prefactorization algebra. As a first step, we will rework the notion of equivariant prefactorization algebra. We will start by constructing a colored operad for each group G. An algebra over this colored operad will recover the preceding definition of a Gequivariant prefactorization algebra. Definition 3.7.2 Let DisjG M denote the colored operad where the set of colors is the set of opens in M and where the operations DisjG M (U1 , . . . , Un | V) are given by the set   (g1 , . . . , gn ) ∈ Gn | ∀i, gi Ui ⊂ V and ∀i = j, gi Ui ∩ gj Uj = ∅ for any choice of inputs U1 , . . . , Un and output V. There is a map of colored operads DisjM → DisjG M , sending each open to itself and sending a nonempty operation U1 , . . . , Un → V to the identity in Gn . Hence, given an algebra A over DisjG M , there is an “underlying” prefactorization algebra on M. Next, observe that for an open set U, the set DisjG M (U | gU) is the coset g · Stab(U), where the stabilizer subgroup Stab(U) ⊂ G consists of elements

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in G that preserve U. Hence for any algebra A over DisjG M , we see that there ∼ =

is an isomorphism σg : A(U) → A(gU) for every g ∈ G and that they compose in the natural way. Hence the “underlying” prefactorization algebra is G-equivariant. Now observe that every operation in DisjG M factors as a collection of unary operations σg arising from the G-action followed by a operation from the “underlying” prefactorization algebra. (If the input opens Ui already sit in the output open V, we are done. Otherwise, pick elements gi of G that move the input opens inside V and keep them pairwise disjoint.) Hence we obtain the following lemma. Lemma 3.7.3 For G a group, every G-equivariant prefactorization algebra on M produces a unique algebra over DisjG M , and conversely. Some notation will make it easier to understand how the G-action intertwines with the structure of the prefactorization algebra A. If (g1 , . . . , gn ) ∈ DisjG M (U1 , . . . , Un | V), then we denote the associated operation by m(g1 ,...,gk ) : A(U1 ) ⊗ · · · ⊗ A(Uk ) → A(V). It can understood as the composition  ⊗σgi  A(Ui ) −−→ A(gi Ui ) → A(V), i

i

where the second map is the structure map of the prefactorization algebra and the first map is given by the unary operations arising from the action of G on M.

3.7.3 Smooth Equivariance From now on, we focus on the situation where G is a Lie group acting smoothly on a smooth manifold M. In this situation, an algebra A over DisjG M has an underlying prefactorization algebra on M but now we want the operations to vary smoothly as the input opens are moved by elements of G. To accomplish this, we need to equip with sets of operations with a “smooth structure” and we need the target category, in which an algebra takes values, to admit a notion of “smoothly varying family of multilinear maps.” Throughout this book, our prefactorization algebras take values in the category DVS of differentiable vector spaces or in the category Ch(DVS) of cochain complexes of differentiable vector spaces. We view these categories as enriched over themselves. In the case of DVS, the self-enrichement is discussed in Section 3.5.6. If V, W are differentiable vector spaces, we denote by HomDVS (V, W) the differentiable vector space of maps from V to W. The key

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feature of HomDVS (V, W) is that its value on a point is DVS(V, W) and, more generally, its value on a manifold X is smooth families over X of maps of differentiable vector spaces from V to W. The self-enrichment of DVS leads, in an obvious way, to a self-enrichment of Ch(DVS). The Colored Operad Equipping each set of operations DisjG M (U1 , . . . , Un | V) with some kind of smooth structure is a little subtle. We might hope such a set, which is a subset of Gn , inherits a manifold structure from Gn , but it is easy to produce examples of opens Ui and V where the set of operations is far from being a submanifold of Gn . We take the following approach instead. Given a subset S ⊂ X of a smooth manifold, there is a set of smooth maps Y → X that factor through S for any smooth manifold Y. In other words, S provides a sheaf of sets on the site Mfld of smooth manifolds, sometimes known as a “generalized manifold.” (See Definition B.2.1 in Appendix B for the site.) Let Shv(Mfld) denote the category of sheaves (of sets). Then every collection of operations DisjG M (U1 , . . . , Un | V) provides such a sheaf, so we naturally obtain a colored operad enriched in “generalized manifolds,” i.e., Shv(Mfld).

M denote the colored operad where the set of colors Definition 3.7.4 Let Disj

G is the set of opens in M and where the operations Disj M (U1 , . . . , Un | V) are the sheaves on Mfld determined by the subset G

n DisjG M (U1 , . . . , Un | V) ⊂ G .

G Let us unpack what this definition means. An algebra F over Disj M in DVS associates to each open U ⊂ M, a differentiable vector space F(U). To each finite collection of opens U1 , . . . , Un , V, we have a map of sheaves

G Disj M (U1 , . . . , Un | V) → HomDVS (F(U1 ), . . . , F(Un ) | F(V)). Thus, for each smooth manifold Y and for each smooth map Y → Gn factoring through DisjG M (U1 , . . . , Un | V), we obtain a section in C∞ (Y, HomDVS (F(U1 ), . . . , F(Un ) | F(V)), which encodes a Y-family of multilinear morphisms from the F(Ui ) to F(V).

G Note that we can evaluate each sheaf Disj M (U1 , . . . , Un | V) at the point ∗ ∈ G Mfld to obtain the underlying set DisjM (U1 , . . . , Un | V). Thus, each algebra

G over Disj M has an underlying G-equivariant prefactorization algebra.

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An algebra over this new colored operad is very close to what we need for our purposes in this book. What’s missing so far is the ability to differentiate the action of G on the prefactorization algebra to obtain an action of the Lie algebra g. Ideally, this action of g would simply exist. Instead, we will put it in by hand, as data, since that suffices for our purposes. After giving our definition, however, we will indicate a condition on an algebra over DisjG M that provides the desired action automatically. Derivations First, we need to introduce the notion of a derivation of a prefactorization algebra on a manifold M. We will construct a differential graded Lie algebra of derivations of any prefactorization algebra. Recall that a derivation of an associative algebra A is a linear map D : A → A that intertwines in an interesting way with the multiplication map: D(m(a, b)) = m(Da, b) + m(a, Db), a relation known as the Leibniz rule. We simply write down the analog for an algebra over the colored operad DisjM taking values in cochain complexes. Definition 3.7.5 A degree k derivation of a prefactorization algebra F is a collection of maps DU : F(U) → F(U) of cohomological degree k for each open subset U ⊂ M, with the property that for any finite collection of pairwise disjoint opens U1 , . . . , Un , all contained in an open V, and an element αi ∈ F(Ui ) for each open, the derivation acts by a Leibniz rule on the structure maps:  1 ,...,Un 1 ,...,Un (α1 , . . . , αn ) = (−1)k(|α1 |+···+|αi−1 |) mU DV mU V V i

× (α1 , . . . , DUi αi , . . . , αn ), where the sign is determined by the usual Koszul rule of signs. Let Derk (F) denote the derivations of degree k. It is easy to verify that the derivations of all degrees Der∗ (F) forms a differential graded Lie algebra. The differential is defined by (dD)U = [dU , DU ], where dU is the differential on F(U). The Lie bracket is defined by [D, D ]U = [DU , DU ]. The concept of derivation allows us to talk about the action of a dg Lie algebra g on a prefactorization algebra F. Such an action is simply a homomorphism of differential graded Lie algebras from g to Der∗ (F).

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The Main Definition We now provide the main definition. As our prefactorization algebra F takes values in the category of differentiable cochain complexes, it makes sense to differentiate an element of F(U) for any open U. Definition 3.7.6 A smoothly G-equivariant prefactorization algebra on M is

G an algebra F over Disj M and an action of the Lie algebra g of G on the underlying prefactorization algebra of F such that for every X ∈ g, every operation

G (g1 , . . . , gn ) ∈ Disj M (U1 , . . . , Un | V), and every 1 ≤ i ≤ n, we have ∂ mg ,...,g (α1 , . . . , αn ) = mg1 ,...,gn (α1 , . . . , X(αi ), . . . , αn ). ∂Xi 1 n ∂ indicates the action of the left-invariant vector field On the left-hand side, ∂X i k on G associated to X in the ith factor of Gk and zero in the remaining factors.

G Remark: In some cases, an algebra A over Disj M should possess a natural action of g. We want to recover how g acts on each value A(U) from the action of G on A. Suppose V is an open such that gU ⊂ V for every g in some neighborhood of the identity in G. Then we can differentiate the structure maps mg : A(U) → A(V) to obtain a map X : A(U) → A(V) for every X ∈ g. If A(U) = limV⊃U A(V), we obtain via this limit, an action of g on A(U). Hence, if we have A(U) = limV⊃U A(V) for every open U, we obtain an action of g on the underlying prefactorization algebra of A. (The compatibility of the G-action with the structure ensures that we get derivations of A.) ♦ Here is an example we will revisit. Example: Let F be a locally constant, smoothly translation invariant factorization algebra on R, valued in vector spaces. Hence, A = F((0, 1)) has the structure of an associative algebra. Being locally constant means that for any two intervals (0, 1) and (t, t + 1), there is an isomorphism F((0, 1)) ∼ = F((t, t + 1)) coming from the isomorphism F((a, b)) → F(R) associated to inclusion of an interval into R. As F is translation invariant, there is another isomorphism F((0, 1)) → F((t, t+1)) for any t ∈ R. Composing these two isomorphism yields an action of the group R on A = F((0, 1)). One can check that this is an action on associative algebras, not just vector spaces. The fact that F is both smoothly translation-invariant and locally constant means that the action of R on A is smooth, and thus it differentiates to an infinitesimal action of the Lie algebra R on A by derivations. The basis element ∂ ∂x of R becomes a derivation H of A, called the Hamiltonian.

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In the case that F is the cohomology of the factorization algebra of observables of the free scalar field theory on R with mass m, we will see in Section 4.3 in Chapter 4 that the algebra A is the Weyl algebra, generated by p, q, h¯ with commutation relation [p, q] = h. ¯ The Hamiltonian is given by ♦ H(a) = 21h¯ [p2 − m2 q2 , a].

PA RT II First examples of field theories and their observables

4 Free Field Theories

4.1 The Divergence Complex of a Measure In this section, we revisit the ideas and constructions from Chapter 2. Recall that in that chapter, we studied the divergence operator associated to a Gaussian measure on a finite-dimensional vector space and then generalized this construction to the infinite-dimensional vector spaces that occur in field theory. We used these ideas to define a vector space H 0 (Obsq ) of quantum observables. Our goal here is to lift these ideas to the level of cochain complexes. We will find a cochain complex Obsq such that H 0 (Obsq ) is the vector space we constructed in Chapter 2. To make the narrative as clear as possible, we will recapitulate our approach there.

4.1.1 The Divergence Complex of a Finite-Dimensional Measure and Its Classical Limit We start by considering again Gaussian integrals in finite dimensions. Let M be a smooth manifold of dimension n, and let ω0 be a smooth measure on M. Let f be a function on M. (For example, M could be a vector space, ω0 the Lebesgue measure and f a quadratic form.) The divergence operator for the measure ef /h¯ ω0 is a map Divh¯ : Vect(M) → C∞ (M) X → h¯ −1 (Xf ) + Divω0 X. One way to describe the divergence operator is to contract with the volume form ef /h¯ ω0 to identify Vect(M) with n−1 (M) and C∞ (M) with n (M). Explicitly, the contraction of a function φ with ef /h¯ ω0 is the volume form φef /h¯ ω0 , and the contraction of a vector field X with ef /h¯ ω0 is the n − 1-form 89

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ιX ef /h¯ ω0 . Under this identification, the divergence operator is simply the de Rham operator from n−1 (M) to n (M). The de Rham operator, of course, is part of the de Rham complex. In a similar way, we can define the divergence complex, as follows. Let PVi (M) = C∞ (M, ∧i TM) denote the space of polyvector fields on M. The divergence complex is the complex Divh¯

Divh¯

· · · → PVi (M) −−→ PV1 (M) −−→ PV0 (M) where the differential Divh¯ : PVi (M) → PVi−1 (M) is defined so that the diagram PVi (M) 

∨ef /h¯ ω0

Divh¯

PVi−1 (M)

∨ef /h¯ ω0

/ n−i (M) 

ddR

/ n−i+1 (M)

commutes. Here we use ∨ef /h¯ ω0 to denote the contraction map; it is the natural extension to polyvector fields of the contractions we defined for functions and vector fields. In summary, after contracting with the volume ef /h¯ ω0 , the divergence complex becomes the de Rham complex. It is easy to check that, as maps from PVi (M) to PVi−1 (M), we have Divh¯ = ∨h¯ −1 df + Divω0 where ∨df is the operator of contracting with the 1-form df . In the h¯ → 0 limit, the dominant term is ∨h¯ −1 df . More precisely, by multiplying the differential by h, ¯ we see that there is a flat family of cochain complexes over the algebra C[h] ¯ with the properties that (i) The family is isomorphic to the divergence complex when h¯ = 0. (ii) At h¯ = 0, the cochain complex is ∨df

∨df

· · · → PV2 (M) −−→ PV1 (M) −−→ PV0 (M). Note that this second cochain complex is a differential graded algebra, which is not the case for the divergence complex.

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The image of the map ∨df : PV1 (M) → PV0 (M) is the ideal cutting out the critical locus. Indeed, this whole complex is the Koszul complex for the equations cutting out the critical locus. This observation leads to the following definition. Definition 4.1.1 The derived critical locus of f is the locally dg ringed space whose underlying manifold is M and whose dg commutative algebra of functions is the complex PV∗ (M) with differential ∨df . Remark: We will call a manifold with a nice sheaf of dg commutative algebras a dg manifold. Since the purpose of this section is motivational, we will not develop a theory of such dg manifolds. We are using the concrete object here as a way to think about the derived geometry of this situation. (More details on derived geometry, from a different point of view, are discussed in Volume 2.) The crucial point is that the dg algebra keeps track of the behavior of df , including higher homological data. ♦ Here is a bit more motivation for our terminology. Let (df ) ⊂ T ∗ M denote the graph of df . The ordinary critical locus of f is the intersection of (df ) with the zero-section M ⊂ T ∗ M. The derived critical locus is defined to be the derived intersection. In derived geometry, functions on derived intersections are defined by derived tensor products: ∞ C∞ (Crith (f )) = C∞ ((df )) ⊗L C∞ (T ∗ M) C (M).

By using a Koszul resolution of C∞ (M) as a module for C∞ (T ∗ M), one finds a quasi-isomorphism of dg commutative algebras between this derived intersection and the complex PV∗ (M) with differential ∨df . In short, we find that the h¯ → 0 limit of the divergence complex is the dg commutative algebra of functions on the derived critical locus of f . An important special case of this relationship is when the function f is zero. In that case, the derived critical locus of f has as functions the algebra PV∗ (M) with zero differential. We view this algebra as the functions on the graded manifold T ∗ [−1]M. The derived critical locus for a general function f can be viewed as a deformation of T ∗ [−1]M obtained by introducing a differential ∨df .

4.1.2 A Different Construction We will define the prefactorization algebra of observables of a free scalar field theory as a divergence complex, just like we defined H 0 of observables to be given by functions modulo divergences in Chapter 2. It turns out that there

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is a slick way to write this prefactorization algebra as a twisted factorization envelope of a certain sheaf of Heisenberg Lie algebras. We will explain this point in a finite-dimensional toy model, and then use the factorization envelope picture to define the prefactorization algebra of observables of the field theory in the next section. Let V be a vector space, and let q : V → R be a quadratic function on V. Let ω0 be the Lebesgue measure on V. We want to understand the divergence complex for the measure eq/h¯ ω0 . The construction is quite general: we do not need to assume that q is nondegenerate. The derived critical locus of the function q is a linear dg manifold. (The Jacobi ideal is generated by linear equations.) Linear dg manifolds are equivalent to cochain complexes: any cochain complex B gives rise to the linear dg manifold whose functions are the symmetric algebra on the dual of B. The derived critical locus of q is described by the cochain complex W given by ∂/∂q

V −−→ V ∗ [−1], ∂ where the differential sends v ∈ V to the linear functional ∂q v = q(v, −). Note that W is equipped with a graded antisymmetric pairing −, − of cohomological degree −1, defined by pairing V and V ∗ . In other words, W has a symplectic pairing of cohomological degree −1. We let

⊕ W, HW = C · h[−1] ¯ where C · h¯ indicates a one-dimensional vector space with basis h. ¯ We give the cochain complex HW a Lie bracket by saying that   [w, w ] = h¯ w, w . Thus, HW is a shifted-symplectic version of the Heisenberg Lie algebra of an ordinary symplectic vector space. Consider the Chevalley–Eilenberg chain complex C∗ HW . This cochain complex is defined to be the symmetric algebra of the underlying graded vector space HW [1] = W[1] ⊕ C · h¯ but equipped with a differential determined by the Lie bracket. Since the pairing on W identifies W[1] = W ∗ , we can identify  C∗ (HW ) = (Sym W ∗ [h], ¯ dCE ) where dCE denotes the differential. Since, as a graded vector space, W = V ⊕ V ∗ [−1], we have a natural identification Cn (HW ) = PVn (V)[h] ¯ where PV∗ (V) refers to polyvector fields on V with polynomial coefficients and where, as before, we place PVn (V) in degree −n.

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Lemma 4.1.2 The differential on C∗ (HW ) is, under this identification, the operator i−1 h¯ Diveq/h¯ ω0 : PVi (V)[h] ¯ → PV (V)[h], ¯

where ω0 is the Lebesgue measure on V, and q is the quadratic function on V used to define the differential on the complex W. Proof The proof is an explicit calculation, which we leave to interested readers. The calculation is facilitated by choosing a basis of V in which we can explicitly write both the divergence operator and the differential on the Chevalley–Eilenberg complex C∗ (HW ). In what follows, we define the prefactorization algebra of observables of a free field theory as a Chevalley–Eilenberg chain complex of a certain Heisenberg Lie algebra, constructed as in this lemma.

4.2 The Prefactorization Algebra of a Free Field Theory In this section, we construct the prefactorization algebra associated to any free field theory. We will concentrate, however, on the free scalar field theory on a Riemannian manifold. We will show that, for one-dimensional manifolds, this prefactorization algebra recovers the familiar Weyl algebra, the algebra of observables for quantum mechanics. In general, we will show how to construct correlation functions of observables of a free field theory and check that these agree with how physicists define correlation functions.

4.2.1 The Classical Observables of the Free Scalar Field We start with a standard example: the free scalar field. Let M be a Riemannian manifold with metric g, so M is equipped with a natural density. We will use this natural density both to integrate functions and also to provide an isomorphism between functions and densities, and we will use this isomorphism implicitly from hereon. The scalar field theory has smooth functions as fields, and we use the notation φ ∈ C∞ (M) for an arbitrary field. The action functional of the theory is  S(φ) = φφ, M

where  is the Laplacian on M. (Normally we will reserve the symbol  for the Batalin–Vilkovisky Laplacian, but that’s not necessary in this section.)

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This functional is not well defined on a field φ unless φφ is integrable, but it is helpful to bear in mind that for classical field theory, what is crucial is the Euler–Lagrange equation, which in this case is φ = 0 and which is thus welldefined on all smooth functions. The action can be viewed here as a device for producing these partial differential equations. If U ⊂ M is an open subset, then the space of solutions to the equation of motion on U is the space of harmonic functions on U. In this book, we always consider the derived space of solutions of the equation of motion. (For more details about the derived philosophy, readers should consult Volume 2.) In this simple situation, the derived space of solutions to the free field equations, on an open subset U ⊂ M, is the two-term complex 

 ∞ ∞ → C (U)[−1] , E (U) = C (U) − where the complex is concentrated in cohomological degrees 0 and 1. (The bracket [−1] denotes “shift up by 1.”) The observables of this classical field theory are simply the functions on this derived space of solutions to the equations of motion. As this derived space is a cochain complex (and hence linear in nature), it is natural to work with the polynomial functions. (One could work with more complicated types of functions, but the polynomial functions provide a concrete and useful collection of observables.) To be explicit, the classical observables are the symmetric algebra on the dual space to the fields. Let’s make this idea precise, using the technology we introduced in Section 3.5 of Chapter 3. The space E (M) has the structure of differentiable cochain complex (essentially, it is a sheaf of vector spaces on the site of smooth manifolds). We define the space of polynomial functions homogeneous of degree n on E (M) to be the space Pn (E (M)) = HomDVS (E (M), . . . , E (M)|R)Sn . In other words, we consider smooth multilinear maps from n copies of E (M), and then we take the Sn -coinvariants. The algebra of all polynomial functions on E (M) is the space P(E (M)) = n Pn (E (M)). As we discussed in Section 3.5 of Chapter 3, we can identify Pn (E (M)) = Dc (M n , (E! )n )Sn as the Sn -coinvariants of the space of compactly supported distributional sections of the bundle (E! )n on M n . In general, if E (M) is sections of a graded bundle E, then E! is E∨ ⊗ Dens. In the case at hand, the bundle E! is two copies of the trivial bundle, one in degree −1 and one in degree 0.

4.2 The Prefactorization Algebra of a Free Field Theory

95

For example, the space P1 (E (M)) = E (M)∨ of smooth linear functionals on E (M) is the space 

∨ −1  0 E (M) = Dc (M) − → Dc (M) , where Dc (M) indicates the space of compactly supported distributions on M. We also want to keep track of where measurements are taking place on the manifold M, so we will organize the observables by where they are supported. The classical observables with support in U ⊂ M are then the symmetric algebra of

  → Dc (U) , E (U)∨ = Dc (U)[1] − where Dc (U) indicates the space of compactly supported distributions on U. The complex E (U)∨ is thus the graded, smooth linear dual to the two-term complex E (U) given earlier. Note that, as the graded dual to E , this complex is concentrated in cohomological degrees 0 and −1. These are precisely the observables that depend only on the behavior of the field φ on the open set U. Thus, as a first pass, one would want to define the classical observables as the symmetric algebra on E (U)∨ . This choice leads, however, to difficulties defining the quantum observables. When we work with an interacting theory, these difficulties can be surmounted only by using the techniques of renormalization. For a free field theory, though, there is a much simpler solution, as we now explain. Recall from Section 3.5 in Chapter 3 that E! denotes the vector bundle E∨ ⊗ DensM . Our identification between densities and functions then produces an isomorphism  Ec! (U) ∼ = Cc∞ (U)[1] → Cc∞ (U) , for compactly supported sections of E ! . Note that there is a natural map of cochain complexes Ec! (U) → E (U)∨ , given by viewing a compactly supported function as a distribution. Lemma 4.2.1 The inclusion map Ec! (U) → E (U)∨ is a cochain homotopy equivalence of differentiable cochain complexes. Proof This assertion is a special case of a general result proved in Appendix D. Note that by differentiable homotopy equivalence we mean that there is an “inverse” map E (U)∨ → Ec! (U), and differentiable cochain homotopies between the two composed maps and the identity maps. “Differentiable” here means that all maps are in the category DVS of differentiable vector spaces.

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As these are convenient cochain complexes, suffices to construct a continuous homotopy equivalence. This lemma says that, since we are working homotopically, we can replace a distributional linear observable by a smooth linear observable. In other words, any distributional observable that is closed in the cochain complex E (U)∨ is chain homotopy equivalent to a closed smooth observable.We think of the smooth linear observables as “smeared.” For example, we can replace a delta function δx by some bump function supported near the point x. The observables we will work with is the space of “smeared observables,” defined by 

→ Cc∞ (U)), Obscl (U) = Sym(Ec! (U)) = Sym(Cc∞ (U)[1] − the symmetric algebra on Ec! (U). As we explained in Section 3.5 in Chapter 3, this symmetric algebra is defined using the natural symmetric monoidal structure on the full subcategory CVS ⊂ DVS of convenient vector spaces. Concretely, we can identify Symn (Ec! (U)) = Cc∞ (U n , [E! ]⊗n )Sn . In other words, Symn Ec! (U) is the subspace of Pn (E (U)) defined by taking all distributions to be smooth functions with compact support. Lemma 4.2.2 The map Obscl (U) → P(E (U)) is a homotopy equivalence of cochain complexes of differentiable vector spaces. Proof It suffices to show that the map Symn Ec! (U) → Pn (E (U)) is a differentiable homotopy equivalence for each n. But for this, it suffices to observe that the map Cc∞ (U, [E! ]⊗n ) → Dc (U, [E! ]⊗n ) is an Sn -equivariant differentiable homotopy equivalence. In parallel to the previous lemma, this lemma says that, since we are working homotopically, we can replace a distributional polynomial observable (given by integration against some distribution on U n ) by a smooth polynomial observable (given by integration against a smooth function on U n ).

4.2.2 Interpreting this Construction Let us describe the cochain complex Obscl (U) more explicitly, to clarify the relationship with what we discussed in Chapter 2. The complex Obscl (U) looks

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97

like · · · → ∧2 Cc∞ (U) ⊗ Sym Cc∞ (U) → Cc∞ (U) ⊗ Sym Cc∞ (U) → Sym Cc∞ (U). All tensor products appearing in this expression are completed tensor products in the category of convenient vector spaces. We should interpret Sym Cc∞ (U) as being an algebra of polynomial functions on C∞ (U), using (as we explained previously) the Riemannian volume form on U to identify Cc∞ (U) with a subspace of the dual of C∞ (U). There is a similar interpretation of the other terms in this complex using the geometry of the space C∞ (U) of fields. Let Tc C∞ (U) refer to the sub-bundle of the tangent bundle of C∞ (U) given by the subspace Cc∞ (U) ⊂ C∞ (U). An element of a fiber of Tc C∞ (U) is a first-order variation of a field that is zero outside of a compact set. This subspace Tc C∞ (U) defines an integrable foliation on C∞ (U), and this foliation can be defined if we replace C∞ (U) by any sheaf of spaces. Then, we can interpret Cc∞ (U) ⊗ Sym Cc∞ (U) as a space of polynomial sections of Tc C∞ (U). Similarly, ∧k Cc∞ (U) ⊗ Sym Cc∞ (U) should be interpreted as a space of polynomial sections of the bundle ∧k Tc C∞ (U). That is, if PVc (C∞ (U)) refers to polynomial polyvector fields on C∞ (U) along the foliation given by Tc C∞ (U), we have Obscl (U) = PVc (C∞ (U)). So far, this is just an identification of graded vector spaces. We need to explain how to identify the differential. Roughly speaking, the differential on Obscl (U) corresponds to the differential on PVc (C∞ (U)) obtain this complex is given by contracting with the 1-form dS for the function  1 S(φ) = 2 φφ, the action functional. Before making this idea precise, we recall the finitedimensional model of a quadratic function Q(x) = (x, Ax) on a vector space V. The contraction of the 1-form dQ with a tangent vector v0 ∈ T0 V gives the linear functional x → (v0 , Ax). As S is quadratic in φ, we expect that the contraction of dS with a tangent vector φ0 inC∞ (U) is the linear functional  1 φ → 2 φ0 φ. But we run into an issue here: the functional S is not well defined for all fields φ, because the integral may not converge, and similarly the linear functional above is not well-defined for arbitrary smooth functions φ0 and φ. However,

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the expression ∂S (φ) = ∂φ0

 1 2

φ0 φ

does make sense for any φ ∈ C∞ (U) and φ0 ∈ Cc∞ (U). (Note that we now only consider tangent vectors φ0 with compact support.) In other words, the desired 1-form dS does not make sense as a section of the cotangent bundle of C∞ (U), but it is well defined as a section of the space Tc∗ C∞ (U), the dual of the subbundle Tc C∞ (U) ⊂ TC∞ (U) describing vector fields along the leaves. This leafwise 1-form is closed. Such 1-forms are the kinds of things we can contract with elements of PVc (C∞ (U)). The differential on PVc (C∞ (U)) thus matches the differential on Obscl given by contracting with dS.

4.2.3 General Free Field Theories With this example in mind, we introduce a general definition. Definition 4.2.3 Let M be a manifold. A free field theory on M is the following data: (i) A graded vector bundle E on M, whose sheaf of sections will be denoted E , and whose compactly supported sections will be denoted Ec . (ii) A differential operator d : E → E , of cohomological degree 1 and square zero, making E into an elliptic complex. (iii) Let E! = E∨ ⊗ DensM , and let E ! be the sections of E! . Let d! be the differential on E ! which is the formal adjoint to the differential on E . Note that there is a natural pairing between Ec (U) and E ! (U), and this pairing is compatible with differentials. We require an isomorphism E → E! [−1] compatible with differentials, with the property that the induced pairing of cohomological degree −1 on each Ec (U) is graded antisymmetric. The complex E (U) is the derived version of the solutions to the equations of motion of the theory on an open subset U. More motivation for this definition is presented in Volume 2. Note that the equations of motion for a free theory are always linear, so that the space of solutions is a vector space. Similarly, the derived space of solutions of the equations of motion of a free field theory is a cochain complex, which is a linear derived stack. The cochain complex E (U) should be thought of as the derived space of solutions to the equations of motion on an open subset U. As we explain in Volume 2, the pairing on Ec (U) arises from the fact that the

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equations of motion of a field theory are not arbitrary differential equations, but describe the critical locus of an action functional. For example, for the free scalar field theory on a manifold M with mass m, we have, as previously, +m2

E = C∞ (U) −−−→ C∞ (U)[−1]. Our convention is that  is a nonnegative operator, so that on Rn ,  2  = − ∂x∂ i . The pairing on Ec (U) is defined by "  ! φ0φ1 φ0, φ1 = M

for φ k in the graded piece C∞ (U)[k]. As another example, let us describe Abelian Yang–Mills theory (with gauge group R) in this language. Let M be a manifold of dimension 4. If A ∈ 1 (M) is a connection on the trivial R-bundle on a manifold M, then the Yang–Mills action functional applied to A is   dA ∧ ∗dA = 12 A(d ∗ d)A. SYM (A) = − 12 M

M

The equations of motion are that d ∗ dA = 0. There is also gauge symmetry, given by X ∈ 0 (M), which acts on A by A → A + dX. The complex E describing this theory is d

d∗d

d

→ 1 (M) −−→ 3 (M)[−1] − → 4 (M)[−2]. E = 0 (M)[1] − We explain how to derive this statement in Volume 2. For now, note that H 0 (E ) is the space of those A ∈ 1 (M) that satisfy the Yang–Mills equation d ∗ dA = 0, modulo gauge symmetry. For any free field theory with cochain complex of fields E , we define the classical observables of the theory as Obscl (U) = Sym(Ec! (U)) = Sym(Ec (U)[1]). It is clear that classical observables form a prefactorization algebra (recall the example in Section 3.1.1). Indeed, Obscl (U) is a commutative differential graded algebra for every open U. If U ⊂ V, there is a natural algebra homomorphism cl cl iU V : Obs (U) → Obs (V),

which on generators is the extension-by-zero map Cc∞ (U) → Cc∞ (V).

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If U1 , . . . , Un ⊂ V are disjoint open subsets, the prefactorization structure map is the continuous multilinear map Obscl (U1 ) × · · · × Obscl (Un ) (α1 , . . . , αn )

→ →

Obscl (V) #n Ui i=1 iV αi .

The product denotes the product in the symmetric algebra on V. On a field φ, this observable takes the value n 

i iU V αi (φ) = α1 (φ|U1 ) · · · αn (φ|Un ),

i=1

which is the product of the value that each observable αi takes on φ restricted to Ui .

4.2.4 The One-Dimensional Case, in Detail This space is particularly simple in dimension 1. Indeed, we recover the usual answer at the level of cohomology. Lemma 4.2.4 If U = (a, b) ⊂ R is an interval in R, then the algebra of classical observables for the free field with mass m ≥ 0 has cohomology H ∗ (Obscl ((a, b))) = R[p, q], the polynomial algebra in two variables. Proof The idea is the following. The equations of motion for free classical mechanics on the interval (a, b) are that the field φ satisfies ( + m2 )φ = 0. This space is two dimensional, spanned by the functions {e±mx }, for m > 0, and by the functions {1, x}, for m = 0. Classical observables are functions on the space of solutions to the equations of motion. We would this expect that classical observables are a polynomial algebra in two generators. We need to be careful, however, because we use the derived version of the space of solutions to the equations of motion. We will show that the complex 

2 ! ∞ −1 +m ∞ 0 Ec ((a, b)) = Cc ((a, b)) −−−→ Cc ((a, b)) is homotopy equivalent to the complex R2 situated in degree 0. Since the algebra Obscl ((a, b)) of observables is defined to be the symmetric algebra on this complex, this will imply the result. Without loss of generality, we can take a = −1 and b = 1.

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First, let us introduce some notation. We start in the complex E of fields. If m = 0, let φq (x) = 1 and φp (x) = x. If m > 0, let  φq (x) = 12 emx + e−mx ,  mx 1 φp (x) = 2m e − e−mx . For any value of m, the functions φp and φq are annihilated by the operator −∂x2 +m2 , and they form a basis for the kernel of this operator. Further, φp (0) = 1 and φq (0) = 0, whereas φp (0) = 0 and φq (0) = 1. Finally, φq = φp . Define a map π : Ec! ((−1, 1)) → R{p, q} by

 g → π(g) = q

 g(x)φq dx + p

g(x)φp dx.

It is a cochain map, because if g = ( + m2 )f , where f has compact support, then π(g) = 0. This map is easily seen to be surjective. This map π says how to identify a linear observable on the field φ over the time interval (−1, 1) into a linear combination of the “position” and “momentum.” We need to construct a contracting homotopy on the kernel of π . That is, if Ker π k ⊂ Cc∞ ((−1, 1)) refers to the kernel of π in cohomological degree k, we need to construct an inverse to the differential +m2

Cc∞ ((−1, 1)) = Ker π −1 −−−→ Ker π 0 . This inverse is defined as follows. Let G(x) ∈ C0 (R) be the Green’s function for the operator  + m2 . Explicitly, we have $ m −m|x| e if m > 0 G(x) = 2 1 if m = 0. − 2 |x| Then ( + m2 )G is the delta function at 0. The inverse map sends a function f ∈ Ker π 0 ⊂ Cc∞ ((−1, 1)) to

 Gf =

G(x − y)f (y) dy. y

The fact that f φq = 0 and f φp = 0 implies that G  f has compact support. The fact that G is the Green’s function implies that this operator is the inverse

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to  + m2 . It is clear that the operator of convolution with G is smooth (and even continuous), so the result follows.

4.2.5 The Poisson Bracket We now return to the general case and construct the P0 algebra structure on classical observables. Suppose we have any free field theory on a manifold M, with complex of fields E . Classical observables are the symmetric algebra Sym Ec (U)[1]. Recall that the complex Ec (U) is equipped with an antisymmetric pairing of cohomological degree −1. Thus, Ec (U)[1] is equipped with a symmetric pairing of degree 1. Lemma 4.2.5 There is a unique smooth Poisson bracket on Obscl (U) of cohomological degree 1, with the property that {α, β} = α, β for any two linear observables α, β ∈ Ec (U)[1]. Recall that “smooth” means that the Poisson bracket is a smooth bilinear map {−, −} : Obscl (U) × Obscl (U) → Obscl (U) as defined in Section 3.5 in Chapter 3. Proof The argument we will give is very general, and it applies in any reasonable symmetric monoidal category. Recall that, as stated in Section 3.5 in Chapter 3, the category of convenient vector spaces is a symmetric monoidal category with internal Homs and a Hom-tensor adjunction. Let A be a commutative algebra object in the category Ch(CVS) of convenient cochain complexes, and let M be a dg A-module. Then we define Der(A, M) to be the space of algebra homomorphisms φ : A → A ⊕ M that are the identity on A modulo the ideal M. (Here A ⊕ M is given the “square-zero” algebra structure, where the product of any two elements in M is zero and the product of an element in A with one in M is via the module structure.) Since the category of convenient cochain complexes has internal Homs, this cochain complex Der(A, M) is again an A-module in Ch(CVS). The commutative algebra Obscl (U) = Sym Ec! (U) is the initial commutative algebra in the category Ch(CVS) of convenient cochain complexes equipped with a smooth linear cochain map Ec! (U) → Obscl (U).

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103

(We are simply stating the universal property characterizing Sym.) It follows that for any dg module M in Ch(CVS) over the algebra Sym Ec! (U), Der(Sym Ec! (U), M) = HomDVS (Ec! (U), M). A Poisson bracket on Sym Ec! (U) is, in particular, a biderivation. A biderivation is something that assigns to an element of Sym Ec! (U) a derivation of the algebra Sym Ec! (U). Thus, the space of biderivations is the space    Der Sym Ec! (U), Der Sym Ec! (U), Sym Ec! (U) . What we have said so far thus identifies the space of biderivations with HomDVS (Ec! (U), HomDVS (Ec! (U), Sym Ec! (U))), or equivalently with HomDVS (Ec! (U) ⊗ Ec! (U), Sym Ec! (U)) via the Hom-tensor adjunction in the category Ch(CVS). The Poisson bracket we are constructing corresponds to the biderivation which is the pairing on Ec! (U) viewed as a map Ec! (U) ⊗ Ec! (U) → R = Sym0 Ec! (U). This biderivation is antisymmetric and satisfies the Jacobi identity. Since Poisson brackets are a subspace of biderivations, we have proved both the existence and uniqueness clauses. Note that for U1 , U2 disjoint open subsets of V and for observables αi ∈ Obscl (Ui ), we have U2 1 {iU V α1 , iV α2 } = 0

in Obscl (V). That is, observables coming from disjoint open subsets commute with respect to the Poisson bracket. This property of Obscl (U) is the definition of a P0 prefactorization algebra. (We will show in Section 6.5 in Chapter 6 that this prefactorization algebra is actually a prefactorization algebra.) In the case of classical observables of the free scalar field theory, we can think of Obscl (U) as the space of polyvector fields on C∞ (U) along the foliation of C∞ (U) given by the subspace Cc∞ (U) ⊂ C∞ (U). The Poisson bracket we have just defined is the Schouten bracket on polyvector fields.

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4.2.6 The Quantum Observables of a Free Field Theory In Chapter 2, we constructed a prefactorization algebra that we called H 0 (Obsq ), the quantum observables of a free scalar field theory on a manifold. This space is defined as a space of functions on the space of fields, modulo the image of a certain divergence operator. The aim of this section is to lift this vector space H 0 (Obsq ) to a cochain complex Obsq . As we explained in Section 4.1, this cochain complex will be the analog of the divergence complex of a measure in finite dimensions. In Section 4.1, we explained that for a quadratic function q on a vector space V, the divergence complex for the measure eq/h¯ ω0 on V (where ω0 is the Lebesgue measure) can be realized as the Chevalley–Eilenberg chain complex of a certain Heisenberg Lie algebra. We follow this procedure in defining Obsq . We construct the prefactorization algebra Obsq (U) as a twisted factorization envelope of a sheaf of Lie algebras. Let Ec (U) = Ec (U) ⊕ R · h¯ where Rh¯ denotes the one-dimensional real vector space situated in degree 1 and spanned by h¯ . We give Ec (U) a Lie bracket by saying that, for α, β ∈ Ec (U), [α, β] = h¯ α, β . Thus, Ec (U) is a graded version of a Heisenberg Lie algebra, centrally extending the Abelian dg Lie algebra Ec (U). Let Obsq (U) = C∗ (Ec (U)), where C∗ denotes the Chevalley–Eilenberg complex for the Lie algebra homolβ on the category of conveogy of Ec (U), defined using the tensor product ⊗ nient vector spaces, as discussed in Section 3.5 in Chapter 3. Thus,   Obsq (U) = Sym Ec (U)[1] , d   = Obscl (U)[h], ¯ d where the differential arises from the Lie bracket and differential on Ec (U). The symbol [1] indicates a shift of degree down by one. (Recall that we always work with cochain complexes, so our grading convention of C∗ is the negative of one common convention.)

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105

Remark: Those readers who are operadically inclined might notice that the Lie algebra chain complex of a Lie algebra g is the E0 version of the universal enveloping algebra of a Lie algebra. Thus, our construction is an E0 version of the familiar construction of the Weyl algebra as a universal enveloping algebra of a Heisenberg algebra. ♦ Since this is an example of the general construction we discussed in Section 3.6 in Chapter 3, we see that Obsq (U) has the structure of a prefactorization algebra. As we discussed in Section 4.2.2, we can view classical observables on an open set U for the free scalar theory as a certain complex of polyvector fields on the space C∞ (U):  Obscl (U) = PVc (C∞ (U)), ∨dS . By PVc (C∞ (U)) we mean polyvector fields along the foliation Tc C∞ (U) ⊂ TC∞ (U) of compactly supported variants of a field. Concretely, the cochain complex Obscl (U) is ∨dS

∨dS

· · · −−→ ∧2 Cc∞ (U) ⊗ Sym Cc∞ (U) −−→ Cc∞ (U) ∨dS

⊗ Sym Cc∞ (U) −−→ Sym Cc∞ (U), β . where all tensor products appearing in this expression are ⊗ In a similar way, the cochain complex of quantum observables Obsq (U) involves just a modification of the differential. It is ∨dS+h¯ Div

∨dS+h¯ Div

. . . −−−−−−→ Cc∞ (U) ⊗ Sym Cc∞ (U) −−−−−−→ Sym Cc∞ (U). The operator Div is the extension to all polyvector fields of the operator (see Definition 2.2.1) defined in Chapter 2 as a map from polynomial vector fields on C∞ (U) to polynomial functions. Thus, H 0 (Obsq (U)) is the same vector space (and same prefactorization algebra) that we defined in Chapter 2. As a graded vector space, there is an isomorphism Obsq (U) = Obscl (U)[h], ¯ but it does not respect the differentials. In particular, the differential d on the quantum observables satisfies (i) Modulo h¯ , d coincides with the differential on Obscl (U), and (ii) the equation d(a · b) = (da) · b + (−1)|a| a · b + (−1)|a| h{a, ¯ b}.

(4.2.6.1)

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Here, · indicates the commutative product on Obscl (U). As we explain in Volume 2, these properties imply that Obsq defines a prefactorization algebra valued in Beilinson–Drinfeld algebras and that Obsq quantizes the prefactorization algebra Obscl valued in P0 algebras. (See Section A.3.2 in Appendix A for the definition of these types of algebras.) It is a characterizing feature of the Batalin–Vilkovisky formalism that the Poisson bracket measures the failure of the differential d to be a derivation, and the language of P0 and BD algebras is an operadic formalization of this concept. We prove in Section 6.5 in Chapter 6 that Obscl is a factorization algebra, which implies Obsq is a factorization algebra over R[h] ¯ .

4.3 Quantum Mechanics and the Weyl Algebra We will now show that our construction of the free scalar field on the real line R recovers the Weyl algebra, which is the associative algebra of observables in quantum mechanics . We already showed in Section 4.2.4 that the classical observables recovers functions on the phase space. First, we must check that this prefactorization algebra is locally constant and so gives us an associative algebra. Lemma 4.3.1 The prefactorization algebra Obsq on R constructed from the free scalar field theory with mass m is locally constant. Proof Recall that Obsq (U) is the Chevalley–Eilenberg chains on the Heisen+m2

berg Lie algebra H(U) built as a central extension of Cc∞ (U) −−−→ Cc∞ (U)[−1]. Let us filter Obsq (U) by saying that F ≤k Obsq (U) = Sym≤k (H(U)[1]). The associated graded for this filtration is Obscl (U)[h]. ¯ Thus, to show that H ∗ Obsq is locally constant, it suffices to show that H ∗ Obscl is locally constant, by considering the spectral sequence associated to this filtration. We have already seen that H ∗ (Obscl (a, b)) = R[p, q] for any interval (a, b), and that the inclusion maps (a, b) → (a , b ) induces isomorphisms. Thus, the cohomology of Obscl is locally constant, as desired. It follows that the cohomology of this prefactorization algebra is an associative algebra, which we call Am . We will show that Am is the Weyl algebra, no matter the mass m. In fact, we will show more, showing that the prefactorization algebra does know the mass m, encoded in a Hamiltonian operator. The prefactorization

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107

algebra for the free scalar field theory on R is built as Chevalley–Eilenberg +m2

chains of the Heisenberg algebra based on Cc∞ (U) −−−→ Cc∞ (U)[−1]. The ∂ of infinitesimal translation acts on Cc∞ (U), commutes with the operator ∂x operator  + m2 , and preserves the cocycle defining the central extension. Therefore, it acts naturally on the Chevalley–Eilenberg chains of the Heisenberg algebra. One can check that this operator is a derivation for the factoriza∂ commutes with inclusions of one open tion product. That is, the operator ∂x subset into another, and if U, V are disjoint and α ∈ Obsq (U), β ∈ Obsq (V), we have ∂ ∂ ∂ (α · β) = (α) · β + α · (β) ∈ Obsq (U  V). ∂x ∂x ∂x In other words, the prefactorization algebra is, in a sense, translation-invariant! (Derivations, and this interpretation, are discussed in more detail in Section 4.8.) ∂ defines a derivation of the associative algebra It follows immediately that ∂x Am coming from the cohomology of Obsq ((0, 1)). ∂ apply to the classical observables, too. Remark: These observations about ∂x Indeed, one discovers that infinitesimal translation induces a derivation of the Poisson algebra R[p, q]. As this derivation preserves the Poisson bracket, we know there is an element H of R[p, q] such that {H, −} is the derivation. Here, H = p2 − m2 q2 . (On a symplectic vector space, every symplectic vector field is Hamiltonian.) ♦

Definition 4.3.2 The Hamiltonian H is the derivation of the associative algebra ∂ of the prefactorization algebra Obsq of Am arising from the derivation − ∂x observables of the free scalar field theory with mass m. Proposition 4.3.3 The associative algebra Am coming from the free scalar field theory with mass m is the Weyl algebra, generated by p, q, h¯ with the relation [p, q] = h¯ and all other commutators being zero. The Hamiltonian H is the derivation H(a) =

2 1 2h¯ [p

− m2 q2 , a].

Note that it is an inner, or principal, derivation. Remark: It might seem a priori that different action functionals, which encode different theories, should have different algebras of observables. In this case, we see that all the theories lead to the same associative algebra. The Hamiltonian is what distinguishes these algebras of observables; that is, different

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actions differ by “how to relate the same observable applied at different moments in time.” There is an important feature of the Weyl algebra that illuminates the role of the Hamiltonian. The Weyl algebra is rigid: the Hochschild cohomology of the Weyl algebra A(n) for a symplectic vector space of dimension 2n vanishes, except in degree 0, where the cohomology is one-dimensional. Hence, in particular, HH 2 (A(n) ) = 0, so there are no nontrivial deformations of the Weyl algebra. In consequence, any action functional whose underlying free theory is the free scalar field (when the coupling constants are formal parameters) will have the same algebra of observables, as an associative algebra. Since HH 1 (A(n) ) = 0, we know that every derivation is inner and hence is represented by some element of A(n) . Thus, the derivation arising from translation must have a Hamiltonian operator. ♦ Proof We start by writing down elements of Obsq corresponding to the position and momentum observables. Recall that   Obsq ((a, b)) = Sym Cc∞ ((a, b))−1 ⊕ Cc∞ ((a, b))0 [h] ¯ with a certain differential. We let φq , φp ∈ C∞ (R) be the functions introduced in the proof of Lemma 4.2.4. Explicitly, if m = 0, then φq (x) = 1 and φp (x) = x, whereas if m = 0 we have  φq (x) = 12 emx + e−mx ,  mx 1 φp (x) = 2m e − e−mx . Thus, φq , φp are both in the null space of the operator −∂x2 + m2 , with the properties that ∂x φp = φq and that φq is symmetric under x → −x, whereas φp is antisymmetric. As in the proof of Lemma 4.2.4, choose a function f ∈ Cc∞ ((− 12 , 12 )) that is symmetric under x → −x and has the property that  ∞ f (x)φq (x) dx = 1. −∞

The symmetry of f implies that the integral of f against φp is zero. Let ft ∈ Cc∞ ((t − 12 , t + 12 )) be ft (x) = f (x − t). We define observables Pt , Qt by Qt = ft , Pt = −ft .

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The observables Qt , Pt are in the space Cc∞ (It )[1]⊕Cc∞ (I) of linear observables for the interval It = ((t − 12 , t + 12 ). They are also of cohomological degree 0. If we think of observables as functionals of a field in C∞ (It ) ⊕ C∞ (It )[−1], then these are linear observables given by integrating ft or −ft against the field φ. Thus, Qt and Pt represent average measurements of positions and momenta of the field φ in a neighborhood of t. Because the cohomology classes [P0 ], [Q0 ] generate the commutative algebra H ∗ (Obscl (R)), it is automatic that they still generate the associative algebra H 0 Obsq (R). We thus need to show that they satisfy the Heisenberg commutation relation [[P0 ], [Q0 ]] = h¯ for the associative product on H 0 Obsq (R), which is an associative algebra by virtue of the fact that H ∗ Obsq is locally constant. (The other commutators among [P0 ] and [Q0 ] vanish automatically.) We also need to calculate the Hamiltonian. We see that ∂ − Qt = −ft = Pt = dtd f (x − t) = dtd Qt , ∂x ∂ − Pt = ft . ∂x Hence, the Hamiltonian acting on [P0 ] gives the t-derivative of [Pt ] at t = 0 and similarly for Q0 . In other words, the parameter t encodes translation of the real line, and the infinitesimal action of translation acts correctly on the observables. In cohomology, the image of −∂x2 + m2 is zero, we see that d dt [Pt ]

= 0 + m2 [f (x − t)] = m2 [Qt ],

as we expect from usual classical mechanics. In particular, when m = 0, [Pt ] is independent of t: in other words, momentum is conserved. Thus, the Hamiltonian H satisfies H([P0 ]) = m2 [Q0 ] H([Q0 ]) = [P0 ]. If we assume that the commutation relation [[P0 ], [Q0 ]] = h¯ holds, then % & H(a) = 21h¯ [P0 ]2 − m2 [Q0 ]2 , a , as desired. Thus, to complete the proof, it only remains to verify the Heisenberg commutation relation.

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If a(t) is a function t, let a˙ (t) denote the t-derivative. It follows from the equations above for the t-derivatives of [Pt ] and [Qt ] that for any function a(t) satisfying a¨ (t) = m2 a(t), the observable a(t)Pt − a˙ (t)Qt is independent of t at the level of cohomology. More precisely, if  Qt is the linear observable of cohomological degree −1 given by the function ft in Cc∞ ((t − 12 , t + 12 )[1], we have ∂ Qt ) (a(t)Pt − a˙ (t)Qt ) = −d(a(t) ∂t where d is the differential on observables. Explicitly, we compute ∂ ∂ ∂ ∂ ∂ ft (x) − a¨ (t)ft (x) − a˙ (t) ft (x) (a(t)Pt − a˙ (t)Qt ) = −˙a(t) ft (x) − a(t) ∂t ∂x ∂t ∂x ∂t ∂2 = a(t) 2 ft (x) − a¨ (t)ft (x) ∂x ∂2 = a(t) 2 ft (x) − m2 a(t)ft (x) ∂x = −d(a(t) Qt ). We will define modified observables Pt , Qt that are independent of t at the cohomological level. We let Pt = φq (t)Pt − φq (t)Qt Qt = φp (t)Pt − φp (t)Qt . Since φp and φq are in the null space of the operator −∂x2 + m2 , the observables Pt , Qt are independent of t at the level of cohomology. In the case m = 0, then φq (t) = 1 so that Pt = Pt . The statement that Pt is independent of t corresponds, in this case, to conservation of momentum. In general, P0 = P0 and Q0 = Q0 . We also have ∂ Pt = −d(φq (t) Qt ), ∂t and similarly for Qt . It follows that if we define a linear degree −1 observable hs,t by  t hs,t = φq (u) Qu (x) du, u=s

then dhs,t = Ps − Pt .

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Note that if |t| > 1, the observables Pt and Qt have disjoint support from the observables P0 and Q0 . Thus, we can use the prefactorization structure map Obsq ((− 12 , 12 )) ⊗ Obsq ((t − 12 , t + 12 )) → Obsq (R) to define a product observable Q0 · Pt ∈ Obsq (R). We will let  denote the associative multiplication on H 0 Obsq (R). We defined this multiplication by [Q0 ]  [P0 ] = [Q0 · Pt ] for t > 1 [P0 ]  [Q0 ] = [Q0 · Pt ] for t < −1. Thus, it remains to show that, for t > 1, [Q0 · Pt ] − [Q0 · P−t ] = h¯ . We will construct an observable whose differential is the difference between the left- and right-hand sides. Consider the observable S = f (x)h−t,t (y) ∈ Cc∞ (R) ⊗ Cc∞ (R)[1], where the functions f and h−t,t were defined earlier. We view h−t,t as being of cohomological degree −1, and f as being of cohomological degree 1. Recall that the differential on Obsq (R) has two terms: one coming from the Laplacian  + m2 mapping Cc∞ (R)−1 to Cc∞ (R)0 , and one arising from the bracket of the Heisenberg Lie algebra. The second term maps   Sym2 Cc∞ (R)−1 ⊕ Cc∞ (R)0 → R h. ¯ Applying this differential to the observable S, we find that  (dS) = f (x)(−∂ 2 + m2 )h−t,t (y) + h¯ h−t,t (x)f (x) dx R  = Q0 · (P−t − Pt ) + h¯ f (x)h−t,t (x). 

Therefore [Q0 Pt ] − [Q0 P−t ] = h¯

f (x)h−t,t (x).

It remains to compute the integral. This integral can be rewritten as  t  ∞ f (x)f (x − u)φq (u) du. u=−t x=−∞

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Note that the answer is automatically independent of t for t sufficiently large, because f (x) is supported near the origin so that f (x)f (x − u) = 0 for u sufficiently large. Thus, we can sent t → ∞. Since f is also symmetric under x → −x, we can replace f (x−u) by f (u−x). We can perform the u-integral by changing coordinates u → u − x, leaving the integrand as f (x)f (u)φq (u + x). Note that   φq (u + x) = 12 em(x+u) + e−m(x+u) . Now, by assumption on f ,   f (x)emx dx = f (x)e−mx dx = 1. It follows that



∞

φq (u + x)f (u) du = φq (x).

u=−∞

We can then perform the remaining x integral desired. Thus, we have proven



φq (x)f (x) dx, which gives 1, as

[[Q0 ], [P0 ]] = h, ¯ as desired.

4.4 Pushforward and Canonical Quantization Consider the free scalar field theory on a manifold of the form N ×R, equipped with the product metric. We assume for simplicity that N is compact. Let Obsq denote the prefactorization algebra of observables of the free scalar field theory with mass m on N × R. Let π : N × R → R be the projection map. There is a pushforward prefactorization algebra π∗ Obsq living on R and defined by (π∗ Obsq )(U) = Obsq (π −1 (U)). (This pushforward construction is a version of compactification in physics.) In this section, we explain how to relate this prefactorization algebra on R to an infinite tensor product of the prefactorization algebras associated to quantum mechanics on R. See Figure 4.1. Let {ei }i∈I be an orthonormal basis of eigenvectors of the operator  + m2 on C∞ (N), where ei has eigenvalue λi . The direct sum i∈I Rei is a dense subspace of C∞ (N).

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113

π −1 ((−1, 1))

N×R

π

(−1, 1)

R

Figure 4.1. The value of π∗ Obsq on (−1, 1) is the value of Obsq on the preimage π −1 ((−1, 1)).

For m ∈ R, let Am denote the cohomology of the prefactorization algebra associated to the free one-dimensional scalar field theory with mass m. Thus, Am is the Weyl algebra generated by p, q, and h¯ with commutator [p, q] = h¯ . The dependence on m appears only through the Hamiltonian. Consider the tensor product of algebras AN = A√λ1 ⊗R[h¯ ] A√λ2 ⊗R[h¯ ] · · · over the entire spectrum {λi }i∈I . (The infinite tensor product is defined to be the colimit of the finite tensor products, where the maps in the colimit use the unit in each algebra. See the example in Section 3.1.1 for more discussion.) Proposition 4.4.1 There is a dense sub-prefactorization algebra of π∗ Obsq that is locally constant and whose cohomology prefactorization algebra corresponds to the associative algebra AN . Remark: This proposition encodes, in essence, the procedure usually known as canonical quantization. (In a physics textbook, N is typically a torus S1 × · · · × S1 whose radii are eventually “sent to infinity.”) The Weyl algebra A√λi for eigenvalue λi corresponds to “a quantum mechanical particle evolving in N with mass m and energy λi .” (An important difference is that a quantum field theory textbook often works with Lorentzian signature, rather than Euclidean signature, as we do.) The time-ordered observables of the whole free scalar field π∗ Obsq are essentially described by the combination of the algebras for all these energy levels. The different energy levels do not affect

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one another, as it is a free theory. When an interaction term is added to the action functional, these energy levels do interact. The prefactorization algebra π∗ Obsq has a derivation, the Hamiltonian, coming from infinitesimal translation in R. The prefactorization algebra A√λi also has a Hamiltonian, given by bracketing with 21h¯ [p2 − λi q2 , −]. The map from  √ q ∗ ♦ i A λi to H (π∗ Obs ) intertwines these derivations. Proof The prefactorization algebra π∗ Obsq on R assigns to an open subset U ⊂ R the Chevalley–Eilenberg chains of a Heisenberg Lie algebra given by a central extension of +m2

Cc∞ (U × N) −−−→ C∞ (U × N)[−1]. A dense subcomplex of these linear observables is  

R +λi Cc∞ (U)ei −−−−→ Cc∞ (U)ei [−1] ,

(†)

i∈I

because the topological vector space of smooth functions is a certain completion of this direct sum of eigenspaces. Let Fi be the prefactorization algebra on R associated to quantum mechanics √ with mass λi . This prefactorization algebra is the envelope of the Heisenberg central extension of R +λi

Cc∞ (U)ei −−−−→ Cc∞ (U)ei [−1]. Note that Fi is a prefactorization algebra in modules over R[h]. ¯ We can define then the tensor product prefactorization algebra F = F1 ⊗R[h¯ ] F2 ⊗R[h¯ ] · · · to be the colimit of the finite tensor products of the Fi under the inclusion maps coming from the unit 1 ∈ Fi (U) for any open subset. Equivalently, we can view this tensor product as being associated to the Heisenberg central extension of the complex (†) above. Because the complex (†) is a dense subspace of the complex whose Heisenberg extension defines π∗ Obsq , we see that there is a map of prefactorization algebras with dense image F → π∗ Obsq . Passing to cohomology, we have a map H ∗ F → H ∗ (π∗ Obsq ).

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As the prefactorization algebra H ∗ (Fi ) corresponds to the Weyl algebra A√λi , we see that H ∗ F corresponds to the algebra AN .

4.5 Abelian Chern–Simons Theory We have discussed the free scalar field in some detail, and we have recovered several aspects of the “usual” story. (Further aspects of the free scalar field occupy much of the rest of this chapter.) Another important example of a free field theory is Abelian Chern–Simons theory, which is a particularly simple example of both a topological field theory and a gauge theory. The theory examined here is the perturbative facet of Chern–Simons theory with gauge group U(1). Let M be an orientable manifold of dimension 3. The space of fields is E = ∗M [1], equipped with the exterior derivative d as its differential. Integration provides a skew-symmetric pairing of degree −1:  α, β = (−1)|β| α ∧ β, M

where α, β live in Ec and each has degree 1 less than its usual degree, due to the shift. (Inside the integral, we simply view them as ordinary differential forms, without the shift.) The action functional of Abelian Chern–Simons theory is then S(α) = α, dα , and the associated equation of motion is that dα = 0. In other words, it picks out flat connections. To be more precise, this complex describes the derived space of solutions to the problem of finding flat connections on the trivial complex line bundle on M, up to gauge equivalence. If we focus on the fields of degree 0 – namely, 1-forms – then a solution is precisely a flat connection. Two solutions α and α  such that α − α  = df , for f ∈ 0 [1], are viewed as equivalent solutions. The function s = ef is a nowhere-vanishing section of the complex line bundle, and we can use it to relate the two connections, as follows. If g is a function, then  s−1 (d + α  )(sg) = s−1 (ds)g + sdg + α  sg = (dg + α  g) + (df )g = (d + α)g, since ds = sdf . In other words, s provides an automorphism, or gauge transformation, intertwining the two flat connections.

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We have explained why 1 appears in degree 0 (the connection 1-forms are the main actors) and why 0 appears in degree 1 (functions are the Lie algebra of gauge automorphisms). We now explain why 2 and 3 appear. The action functional S is naturally a function on this linear derived space d

0 [1] → 1 , so that its exterior derivative dS is a section of the cotangent bundle of this linear derived space. As S is quadratic, dS is linear, so the section is linear. From the physical perspective, we want to study the derived intersection of the zero section and dS in this cotangent bundle: this amounts to studying the derived critical locus of S (i.e., the points in the linear derived stack where dS vanishes). This derived intersection is itself a linear derived stack and is described by the full de Rham complex.

4.5.1 Observables The classical observables for Abelian Chern–Simons theory are easy to describe and to compute. For U an open subset of M, we have   d d d → 1c [1] − → 2c − → 3c [−1] . Obscl (U) = Sym∗ 0c [2] − Thus, we see that  H ∗ Obscl (U) ∼ = Sym∗ Hc∗ (U)[2] . In contrast to the free scalar field, where the cohomology of the linear observables was often infinite-dimensional, the cohomology of the linear observables is finite-dimensional, for any open U. Thus, the cohomology prefactorization algebra H ∗ Obscl is relatively easy to understand. This pleasant situation continues with the quantum observables. Consider the natural filtration on Obsq (U) by  F k Obsq (U) = Sym≤k ∗c (U)[2] [h]. ¯ This filtration induces a spectral sequence, and the first page is H ∗ Obscl (U)[h], ¯ which we have already computed. The differential on the first page is zero, but the differential on the second page arises from the pairing on linear observables (recall that the pairing provides the Lie bracket of the Heisenberg Lie algebra, and this bracket provides the differential in the Chevalley–Eilenberg chain complex). Note that the pairing on linear observables descends to the cohomology of the linear observables by [α], [β] H ∗ = α, β .

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Hence, the differential d2 on the second page of the spectral sequence vanishes on constant and linear terms (i.e., from Sym0 and Sym1 ) and satisfies d2 ([α][β]) = h¯ [α], [β] H ∗ for a pure quadratic term. The BD algebra axiom (recall Eq. (4.2.6.1)) then determines d2 on terms that are cubic and higher. Thus, the cohomology of the quantum observables reduces to understanding the integration pairing between cohomology classes of compactly supported de Rham forms. In particular, Poincar´e duality implies the following. Lemma 4.5.1 For M a connected, closed 3-manifold, 2 H ∗ Obsq (M) ∼ = R[h][1 ¯ − dim H (M)].

Proof Let b2 = dim H 2 (M). Pick a basis α1 , . . . , αb2 for H 2 (M) and a dual basis β1 , . . . , βb2 for H 1 (M): αj , βk H ∗ = δjk . Let ν denote the generator for H 0 (M) and [M] the dual generator for H 3 (M). Thus, d2 (αj βk ) = hδ ¯ jk and d2 (ν[M]) = h, ¯ and d2 vanishes on all other quadratic monomials. Explicit computation then shows that the pure odd term να1 · · · αb2 is closed but not exact, and it generates the cohomology.

4.5.2 Compactifying Along a Closed Surface The next case to consider is a 3-manifold of the form M = R × , where  is a closed, orientable surface. Let π : M → R denote the obvious projection map. We will show that the pushforward prefactorization algebra π∗ Obsq is locally constant, so that it corresponds to an associative algebra. This associative algebra is the Weyl algebra for the graded symplectic vector space H ∗ ()[1]. (Recall Figure 4.1.) Some notation will make it easier to be precise. Let g denote the genus of . Pick a symplectic basis {α1 , . . . , αg , β1 , . . . , βg } for H 1 () in the sense that  αj ∧ βk = δjk . 

Let ν denote the basis for H 0 () given by the constant function 1. Let μ denote a basis for H 3 (M) such that  μ = 1. Thus, the graded vector space H ∗ ()[1] has a natural symplectic structure given by the integration pairing. The odd elements ν and μ are dual under this pairing.

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Proposition 4.5.2 The prefactorization algebra H ∗ π∗ Obsq is locally constant, and it corresponds to the Weyl algebra A for H ∗ ()[1]. In terms of our chosen generators, this Weyl algebra has commutators [ν, μ] = h, ¯ [αj , βk ] = hδ ¯ jk and all others vanish. The Hamiltonian in this Weyl algebra is zero. The fact that the Hamiltonian is zero is what makes Abelian Chern–Simons a topological field theory . Note as well the appearance of the Weyl algebra. Proof This argument is much simpler than the argument for the free scalar field in one dimension. We will use the Hodge theorem to replace the big complex ∗c (R) ⊗ ∗ () by the smaller complex ∗c (R) ⊗ H ∗ (), and then exploit our construction for the universal enveloping algebra (recall Proposition 3.4.1) to obtain the result. Let L = π∗ ∗M [1]. This is the pushforward of the local Lie algebra of linear observables (it is an Abelian Lie algebra, as we are working with a free the denote the pushforward of the Heisenberg ory). Then π∗ Obscl ∼ = C∗ Lc . Let L q ∼ c , the factorization envelope. central extension. Then π∗ Obs = C∗ L By the Hodge theorem, we can obtain a homotopy equivalence between ∗ () and its cohomology H ∗ (). Tensoring with ∗c on R, we obtain a homotopy equivalence Łc " ∗c ⊗ H ∗ ()[1] = g of Abelian dg Lie algebras.Thus, we see that C∗ g is homotopy equivalent to g denotes π∗ Obscl . Similarly, C∗ g is homotopy equivalent to π∗ Obsq , where  the Heisenberg central extension (i.e., we use the Poincar´e pairing on H ∗ () tensored with the integration pairing on R). The cohomology prefactorization algebra H ∗ π∗ Obsq is isomorphic to the cohomology prefactorization algebra H ∗ (C∗ g). By Proposition 3.4.1, we know ()[1], where the central elethis corresponds to the associative algebra U H ∗ ment is denoted h. ¯ Now we compute the Hamiltonian. Recall that the Hamiltonian arose in the free scalar field via the translation action on R. In the text that follows, we will explicitly compute the Hamiltonian in a manner parallel to Proposition 4.3.3. But there is a conceptual reason for the vanishing of the Hamiltonian: the theory, because it is simply the de Rham complex, is homotopy-invariant under diffeomorphism. Not only are the observables preserved by translation along R; they are preserved under any diffeomorphism of R. ∂ preserves everyLet x denote a coordinate on R. Infinitesimal translation ∂x q thing in the prefactorization algebra π∗ Obs , just as in Proposition 4.3.3. In

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this case, however, the solutions to the equations of motion – namely, de Rham cohomology classes of  – are translation-invariant, so the Hamiltonian is trivial, as follows. Fix a function f ∈ Cc∞ ((− 12 , 12 )) such that R f (x)dx = 1. Hence the 1-form f (x)dx generates Hc1 (R). Let ft (x) = f (x − t). Then we get linear observables in C∗ g((t − 12 , t + 12 )) by At,j = ft (x) dx ⊗ αj , Bt,k = ft (x) dx ⊗ βk , νt = ft (x) dx ⊗ ν, μt = ft (x) dx ⊗ μ. These elements are all cohomologically nontrivial, and their cohomology classes generate the Weyl algebra. Observe that, for instance, ∂ At,j = ft (x) dx ⊗ αj = dft ⊗ αj , ∂x so that the infinitesimal translation is cohomologically trivial. This argument applies to all the generators, so we see that the derivation on the Weyl algebra ∂ must be trivial. induced from ∂x Remark: There is a closely related free field theory on M = R × . Equip  with a complex structure, and consider the free theory on M whose fields are ∗ (R) ⊗ 0,∗ ()[1] ⊕ ∗ (R) ⊗ 1,∗ (), concentrated from degrees −1 to 2. (Just to pin down our conventions: 0 (R)⊗ 0,0 () is in degree −1, and 0 (R) ⊗ 1,0 () is in degree 0.) This theory is topological along R and holomorphic along  (see Chapter 5 for more discussion of this notion). Notice that if we added the differential ∂ for  to this complex, we would recover Abelian Chern–Simons on R × . q Let us denote the observables for this theory by Obs∂ . It is straightforward to q mimic the proof above to show that the cohomology of π∗ Obs∂ corresponds to the Weyl algebra on the graded symplectic vector space H ∗,∗ ()[1], the Dolbeault cohomology of . As this cohomology is isomorphic to the de Rham cohomology of , we see that the cohomology prefactorization algeq ♦ bras H ∗ π∗ Obsq and H ∗ π∗ Obs∂ are isomorphic.

4.5.3 3-Manifolds with Boundary We now examine the observables on a compact, connected, orientable 3-manifold M with connected boundary ∂M =  of genus g. Let M denote the open interior, so M = M − ∂M. Speaking loosely, we can view M as a

120

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M 

p

Figure 4.2. Everything right of the dotted copy of  maps to the right endpoint.

“module” over the “algebra”  × R, because we can keep stretching the end of M to include a copy of  × R. We will show that this picture leads to a precise statement about the observables: the cohomology H ∗ Obsq (M) is a module for the Weyl algebra A associated to H ∗ ()[1]. Pick a parametrization of a collar neighborhood N of the boundary ∂M such ∼ =

that we have a diffeomorphism ρ : N − →  × [0, 2). Let π : N → [0, 2) denote composition with the projection map. Let N  = π −1 ((0, 1)) denote a neighborhood of the end of M. Consider the map p : M → (0, 1] where  p(x) =

π(x), 1,

x ∈ π −1 ((0, 1)) , else

which collapses everything in M outside of N  down to a point. By construction, p−1 ((a, 1]) is diffeomorphic to M for any 0 < a. Likewise, p−1 ((a, b)) is diffeomorphic to  × (a, b) for any 0 < a < b < 1. See Figure 4.2. It is clear that p∗ Obsq on the interval (0, 1) is simply the observables compactified along a surface, as in Proposition 4.5.2. To any open set of the form (a, 1] in [0, 1], we see that p∗ Obsq ((a, 1]) is quasi-isomorphic to Obsq (M). Proposition 4.5.3 The cohomology prefactorization algebra H ∗ p∗ Obsq is constructible with respect to the stratification (0, 1] = (0, 1) ∪ {1}. In particular,

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(1) On the stratum (0, 1), H ∗ p∗ Obsq corresponds to the Weyl algebra A . (2) For any open (a, 1], H ∗ p∗ Obsq ((a, 1]) ∼ = H ∗ (Obsq (M)). (3) The structure maps equip H ∗ (Obsq (M)) with the structure of a left A module, whose annihilator is the ideal generated by the Lagrangian subspace Im i∗ ⊂ H ∗ ()[1] for the map i∗ : H ∗ (M; )[1] → H ∗ ()[1] arising from i :  → M. Remark: This proposition – more accurately, its proof – indicates how to construct a topological field theory, in the functorial sense, from the bordism category of oriented 2-manifolds with oriented cobordisms as morphisms to the category of graded algebras with bimodules as morphisms. To a surface , we assign the Weyl algebra A . To a 3-manifold, we associate to the bimodule over Weyl algebras (extending the proposition above to more general 3manifolds with boundary). The final step in constructing the functor – verifying composition – relies on the gluing properties of prefactorization algebras, which we discuss in Chapter 6, although one could verify it directly by computation in this case. ♦ Proof We have already explained claims (1) and (2), so we tackle (3). It is manifest that H ∗ (Obsq (M)) is a left A module, because p∗ Obsq is a prefactorization algebra. Thus, it remains to describe the annihilator of this module. Recall the “half lives, half dies” principle for oriented 3-manifolds: for M a compact, orientable 3-manifold with boundary i : ∂M → M, the image of the map i∗ : H 1 (M) → H 1 (∂M) has dimension half that of H 1 (∂M). (For a proof, see lemma 3.5 of Hatcher (2007).) The image L = Im i∗ is, in fact, a Lagrangian subspace of H 1 (∂M)[1]. Now consider our situation, where M = M − ∂M and ∂M = . We know that the neighborhood N  = p−1 ((0, 1)) of the end of M is diffeomorphic to  × (0, 1). Let j : N  → M denote the inclusion. By Poincar´e duality for compactly supported de Rham cohomology, we know Hc2 (N  ) ∼ = H 1 () and 2 1 ∼ Hc (M) = H (M). Hence, the “half lives, half dies” principle shows that for the extension-by-zero map j! : Hc2 (N  ) → Hc2 (M), the kernel Ker j! is a Lagrangian subspace. We also see that j! : Hc3 (N  ) ∼ = Hc3 (M) and j! : Hc1 (N  ) → Hc1 (M) is the zero map. Thus, the graded subspace Ker j! ⊂ Hc∗ (N  ) is Lagrangian. For the observables, the map j! describes how the linear observables on N  behave as linear observables on M. The observables in Ker j! thus vanish on M, and so the ideal they generate in A vanishes on the module H ∗ Obsq (M). We need to show this ideal is the entire annihilator. Note that the action of A is governed by the integration pairing in Hc∗ (M), and the extension-by-zero map

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Hc∗ (N  ) → Hc∗ (M) respects the integration pairing, so there are no nontrivial relations involving nonlinear observables.

4.5.4 Knots and Links Chern–Simons theory is famous for its relationship with knot theory, notably the Vassiliev invariants. Here we will explain a classic example: how the Gauss linking number appears in Abelian Chern–Simons theory. For simplicity, we work in R3 . Let K : S1 → R3 and K  : S1 → R3 be two disjoint knots. The linking number (K, K  ) of K and K  is the degree of the Gauss map GK,K  : S1 ×S1 → S2 that sends a pair of points (x, y) ∈ K × K  to the unit vector (x − y)/|x − y|. One way to compute it – indeed, the way Gauss first introduced it – is to pull back the standard volume form on S2 and integrate over K × K  . Another way is to pick an oriented disc whose boundary is K  and to count, with signs, the number of times K intersects D (we may wiggle D gently to make it transverse to K). To relate this notion to Chern–Simons theory, we need to find an observable associated to a knot. The central object of Chern–Simons theory is a 1-form A, which we view as describing a connection d + A for the trivial line bundle on R3 . Thus, the natural observable for the knot K : S1 → R3 is  OK (A) = A, K

where we have fixed an orientation on S1 . Note that this operator is linear in A. Remark: The Wilson loop observable WK is exp(OK ), which is the holonomy of the connection d + A around K. We do not work with it here because we want to work with the symmetric algebra on linear observables, rather than the completed symmetric algebra. In the non-Abelian setting, we do work with the completed symmetric algebra, so the analogous Wilson loop observable lives in our quantum observables for non-Abelian Chern–Simons theory. ♦ There is a minor issue with this observable, however: it is distributional in nature, as its support sits on the knot K. It is easy to find a smooth (or “smeared”) observable that captures the same information. (We know such a smooth observable exists abstractly by Lemma 4.2.1, but here we will give an explicit representative.) The key is again to use the pushforward. Consider the 3-manifold T = S1 ×R2 . We view the core C = S1 ×(0, 0) ⊂ T as the “distinguished” knot in T. Any embedding κ : T → R3 induces a pushforward of observables so that we can understand the observables supported in

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tubular neighborhood κ(T) of the knot κ(C) by understanding the observables on T. We now explain how to find a smeared analog of OC , where C is the distinguished knot, in Obsq (T). Fix an orientation of T and an orientation of the embedded circle C ⊂ T. Let μ be a compactly supported volume form on R2 whose support contains the origin and whose total integral is 1. Consider its pullback μT = π ∗ μ along the projection π : T → R2 . Using Cartesian coordinates r, z on R2 and coordinates θ , r, z on T, we have μT = φ(r, z) dr ∧ dz, where φ is a nonnegative bump function with support at the origin (0, 0). Note that μT is independent of θ . We thus have a linear function, or observable,  A ∧ μT Oμ : A → T

on any 1-form A. Using the coordinates from the preceding, we see  φ(x, y)OS1 ×(r,z) (A) dr dz, Oμ (A) = R2

so that Oμ is just a smoothed version of OC . Let’s return to the linking number. Given two knots K and K  , we can find thickened embeddings κ : T → R3 and κ  : T → R3 whose images are disjoint and where κ(C) = K and κ  (C) = K  . Let μκ denote the image of μT in Obsq (κ(T)) under the pushforward by κ, and likewise for μκ  . This observable μκ is a smeared version of OK . ∼ R[h], Lemma 4.5.4 In H 0 (Obsq (R3 )) = ¯ the cohomology class [μκ μκ  ] of the K  ). product observable μκ μκ  is equal to h(K, ¯ Proof We outline here one approach, in the spirit of our preceding work. Below, in the example in Section 4.6.3 we explain the standard proof. The Poincar´e lemma for compactly supported forms tells us that Hc2 (R3 ) = 0. As μκ  is closed, we know there is some 1-form ρ such that dρ = μκ  . The product observable μκ ρ then has cohomological degree −1, and by construction we have   μκ ∧ ρ. (d + h)μ ¯ ¯ κ ρ = μκ μκ + h R3

Thus we need to verify that, up to a sign, the integral agrees with the linking number. We do this by making a good choice of ρ. There is a smooth embedding of a closed 2-disc D → R3 whose oriented boundary is K  . Extend this embedding to a thickened embedding i : R3 → R3 , where D sits inside the input copy of R3 as D × 0 and where this thickened embedding contains the image of κ  . Then i∗ μκ  is cohomologically trivial, by the Poincar´e lemma. Pick a 1-form

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ρ such that dρ = i∗ μκ  . Note that ρ has support in i(R3 ). The product μκ ∧ ρ is smeared version, under Poincar´e duality, of the intersection of i(D) with K. Thus its integral counts the signed intersection.

4.6 Another Take on Quantizing Classical Observables We have given an abstract definition of the prefactorization algebra of quantum observables of a free field theory as the factorization envelope of a certain Heisenberg dg Lie algebra. Our goal in this section is to provide another useful description. The prefactorization algebra of quantum observables Obsq , viewed as a graded prefactorization algebra with no differential, coincides with Obscl [h]. ¯ In particular, the structure maps are the same. The only difference between Obsq and Obscl [h] ¯ is in the differential. In deformation quantization, we deform, however, the product of observables. We saw in Section 4.3 that taking cohomology, the structure maps for the quantum observables do change and correspond to the Weyl algebra. The change in differential modifies the structure maps at the level of cohomology. However, it is possible to approach quantization differently and deform the structure maps at the cochain level, instead of the differential. Hence, in this section we will give an alternative, but equivalent, description of Obsq . We will construct an isomorphism of precosheaves Obsq ∼ = Obscl [h] ¯ that is compatible with differentials. This isomorphism is not compatible, however, with the factorization product. Thus, this isomorphism induces a deformed factorization product on Obscl [h] ¯ corresponding to the factorization product on Obsq . In other words, instead of viewing Obsq as being obtained from Obscl by keeping the factorization product fixed but deforming the differential, we will show that it can be obtained from Obscl by keeping the differential fixed but deforming the product. One advantage of this alternative description is that it is easier to construct correlation functions and vacua in this language. We will now develop these ideas in the case of the free scalar field.

4.6.1 Green’s Functions The isomorphism we will construct between the cochain complexes of quantum and classical observables relies on a Green’s function for the Laplacian.

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Definition 4.6.1 A Green’s function is a distribution G on M × M, preserved under reflection across the diagonal (i.e., symmetric) and satisfying ( ⊗ 1)G = δ , where δ is the δ-distribution on the diagonal M → M × M. A Green’s function for the Laplacian with mass satisfies the equation ( ⊗ 1)G + m2 G = δ . (The convention is that  has positive eigenvalues, so that on Rn ,  =  2 − ∂x∂ i .) If M is compact, then there is no Green’s function for the Laplacian. Instead, there is a unique function  G satisfying ( ⊗ 1) G = δ − 1. However, if we introduce a nonzero mass term, then the operator  + m2 on C∞ (M) is an isomorphism, so that there is a unique Green’s function. If M is noncompact, then there can be a Green’s function for the Laplacian without mass term. For example, if M is Rn , then a choice of Green’s function is $ 1 (d/2 − 1) |x − y|2−n if n = 2 G(x, y) = 4π d/2 1 − 2π log |x − y| if n = 2. As the reader will see, the role of the Green’s function is to produce an isomorphism, so different choices lead to different isomorphisms. These choices do not change the quantum observables themselves, merely how one “promotes” a classical observable to a quantum one.

4.6.2 The Isomorphism of Graded Vector Spaces Let us now turn to the construction of the isomorphism of graded vector spaces between Obscl (U) and Obsq (U) in the presence of a Green’s function. The underlying graded vector space of Obsq (U) is Sym(Cc∞ (U)[1] ⊕ Cc∞ (U))[h]. ¯ In general, for any vector space V, each element P ∈ (V ∨ )⊗2 defines a differential operator ∂P of order two on Sym V, viewed as a commutative algebra, where ∂P is uniquely characterized by the conditions that it is zero on Sym≤1 V and that on Sym2 V it is given by contraction with P. The same holds when we define the symmetric algebra using the completed tensor product.

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In the same way, for every distribution P on U × U, we can define a continuous, second-order differential operator on Sym(Cc∞ (U)[1] ⊕ Cc∞ (U)) that is uniquely characterized by the properties that it vanishes on Sym≤1 , that it vanishes elements of negative cohomological degree in Sym2 (Cc∞ (U)[1] ⊕ Cc∞ (U)), and that  ∂P (φψ) =

P(x, y)φ(x)ψ(y) U×U

for any φ, ψ ∈ Cc∞ (U) of cohomological degree zero. (On the left-hand side, we view φψ as an element of Sym2 .) Choose a Green’s function G for the Laplacian on M. Thus, G restricts to a Green’s function for the Laplacian on any open subset U of M. Therefore, we can define a second-order differential operator ∂G on the algeto bra Sym(Cc∞ (U)[1] ⊕ Cc∞ (U)). We extend this operator by R[h]-linearity ¯ q an operator on the graded vector space Obs (U). Now, the differential on Obsq (U) can be written as d = d1 +d2 , where d1 is a first-order differential operator and d2 is a second-order operator. The operator 

→ d1 is the derivation arising from the differential on the complex Cc∞ (U) − Cc∞ (U). The operator d2 arises from the Lie bracket on the Heisenberg dg Lie algebra; it is a continuous, h-linear, second-order differential operator uniquely ¯ characterized by the property that d2 (φ −1 φ 0 ) = h¯



φ −1 (x)φ 0 (x)

M

for φ k in the copy of Cc∞ (U) in degree k, sitting inside of Obsq (U). The Green’s function G satisfies (( + m2 ) ⊗ 1)G = δ where δ is the Green’s function on the diagonal. It follows that [h∂ ¯ G , d1 ] = d2 . Indeed, both sides of this equation are second-order differential operators, so to check the equation, it suffices to calculate how the act on an element of Sym2 (Cc∞ (U)[1] ⊕ Cc∞ (U)). If φ k denotes an element of the copy of Cc∞ (U)

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in cohomological degree k, we have 0 2 −1 h¯ ∂G d1 (φ 0 φ −1 ) = h∂ ¯ G (φ ( + m )φ )  = G(x, y)φ 0 (x)(( + m2 )φ −1 )(y) U×U ((y + m2 )G(x, y))φ 0 (x)φ −1 (y) = U×U  φ 0 (x)φ −1 (x) = U

= d2 (φ 0 φ −1 ). On the first line, the element φ 0 φ −1 lives in Sym2 , as does its image under d1 . On the fourth line, we have used the fact that y + m2 applied to G(x, y) is the delta-distribution on the diagonal. It is also immediate that ∂G commutes with d2 . Thus, if we define W(α) = h ∂ ¯ e G (α), we have (d1 + d2 )W(α) = W(d1 α). In other words, there is an R[h]-linear cochain isomorphism ¯ q WU : Obscl (U)[h] ¯ → Obs (U),

where the domain is the complex Obscl (U)[h] ¯ with differential d1 and the codomain is the same graded vector space with differential d1 + d2 . This isomorphism holds for every open U. Note that W is not a map of prefactorization algebras. Thus, W induces a factorization product (i.e., structure maps) on classical observables that quantizes the original factorization product. Let us denote this quantum product by h¯ , whereas the original product on classical observables will be denoted by ·. If U, V are disjoint open subsets of M with α ∈ Obscl (U), β ∈ Obscl (V), we have     α h¯ β = e−h¯ ∂G eh¯ ∂G α · eh¯ ∂G β . This product is an analog of the Moyal formula for the product on the Weyl algebra.

4.6.3 Beyond the Free Scalar Field We have described this construction for the case of a free scalar field theory. This construction can be readily generalized to the case of an arbitrary free theory. Suppose we have such a theory on a manifold M, with space of fields

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E (M) and differential d. Instead of a Green’s function, we require a symmetric and continuous linear operator G : (E (M)[1])⊗2 → R such that G d(e1 ⊗ e2 ) = e1 , e2 where −, − is the pairing on E (M), which is part of the data of a free field theory. (When M is compact and H ∗ (E (M)) = 0, the propagator of the theory satisfies this property. For M = Rn , we can generally construct such a G from the Green’s function for the Laplacian.) In this context, the operator eh¯ ∂G is, in the terminology of Costello (2011b), the renormalization group flow operator from scale zero to scale ∞. This construction often makes it easier to analyze the structure of the quantum observables. We will use it, for instance, to understand the correlation functions of some chiral conformal field theories in the next chapter. Example: We can also use it to provide another proof of Lemma 4.5.4, which asserts that Abelian Chern–Simons knows about linking number. Recall that for two disjoint knots K and K  , we introduced smoothed versions, μκ and μκ  , of the currents with support along the knots. It is easy to check that, up to a constant, the 2-form 3  1  (−1)i (xi − yi ) d(x1 − y1 ) ∧ d(x ω= i − yi ) ∧ d(x3 − y3 ) |x − y|3 i=1

R3

R3

on × acts as a Green’s function for the de Rham complex on R3 . It is the pullback of the standard volume form on S2 along the projection (x, y) → (x − y)/|x − y| from the configuration space of two distinct points in R3 to S2 . (Indeed, ω is the standard propagator, as discussed in Costello (2007a) or Kontsevich (1994).) Using the h¯ formula from given earlier, one can see that the cohomology class of a product observable μκ μκ  is precisely the integral formula for the Gauss linking number. To be explicit, we see that    μκ (x)μκ  (y)ω(x, y). μκ h¯ μκ = μκ μκ + h¯ x,y∈R3

Our tubular embeddings, κ : T → R3 and κ  : T → R3 , allow us to rewrite the integral as an integral over T × T ∼ = C × C × R4 . The product C × C of  two circles parametrize K × K , and the last four coordinates parametrize small shifts of K × K  within their tubular neighborhoods. Fixing such a shift, we are integrating ω over K × K  , and hence obtain the linking number (K, K  )

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because ω is a pullback of the volume form on S2 . When we integrate over the shifts, we are integrating μT (x) ∧ μT (y), which integrates to 1 by construction. (We have not checked the overall sign, of course.) ♦

4.7 Correlation Functions Suppose we have a free field theory on a compact manifold M, with the property that H ∗ (E (M)) = 0. As an example, consider the massive scalar field theory on M where +m2

∞ ∞ E = CM −−−→ CM .

Our conventions are such that the eigenvalues of  are nonnegative. Adding a nonzero mass term m2 gives an operator with no zero eigenvalues, so that this complex has no cohomology, by Hodge theory. In this situation, the dg Lie algebra E(M) we constructed above has cohomology spanned by the central element h¯ in degree 1. It follows that there is an isomorphism of R[h]-modules ¯ H ∗ (Obsq (M)) ∼ = R[h]. ¯ Let us normalize this isomorphism by asking that the element 1 in Sym0 (E[1]) = R ⊂ C∗ (E[1]) is sent to 1 ∈ R[h]. ¯ Definition 4.7.1 In this situation, we define the correlation functions of the free theory as follows. For a finite collection U1 , . . . , Un ⊂ M of disjoint opens and a closed element Oi ∈ Obsq (Ui ) for each open, we define O1 · · · On = [O1 · · · On ] ∈ H ∗ (Obsq (M)) = R[h]. ¯ Here O1 · · · On ∈ Obsq (M) denotes the image under the structure map  Obsq (Ui ) → Obsq (M) i

of the tensor product O1 ⊗ · · · ⊗ On . In other words, our normalization gives a value in R[h] ¯ for each cohomology class, and we use this to read off the “expected value” of the product observable.

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The map W constructed in the previous section allows us to calculate correlation functions. Since we have a nonzero mass term and M is compact, there is a unique Green’s function for the operator  + m2 . Lemma 4.7.2 (Wick’s lemma) Let U1 , . . . , Un be pairwise disjoint open sets. Let αi ∈ Obscl (Ui ) = Sym(Cc∞ (Ui )[1] ⊕ Cc∞ (Ui )) be classical observables, and let W(αi ) = eh¯ ∂G αi ∈ Obsq (Ui ) be the corresponding quantum observables under the isomorphism W of cochain complexes. Then W(α1 ) · · · W(αn ) = W −1 (W(α1 ) · · · W(αn )) (0) ∈ R[h]. ¯ On the right-hand side, W(α1 ) · W(α2 ) indicates the product in the algebra Sym(C∞ (M)[1] ⊕ C∞ (M))[h]. ¯ The symbol (0) indicates evaluating a function at zero, i.e., taking its component in Sym0 . on C∞ (M)[1] ⊕ C∞ (M)[h] ¯ Proof The map W is an isomorphism of cochain complexes between Obsq (U) and Obscl (U)[h] ¯ for every open subset U of M. As above, let h¯ denote the factorization product on Obscl [h] ¯ corresponding, under the isomorphism W, to the factorization product on Obsq . That is, we apply W to each input, multiply in Obsq , and then apply W −1 to the output: explicitly, we have   α1 h¯ · · · h¯ αn = e−h¯ ∂G (eh¯ ∂G α1 ) · · · (eh¯ ∂G αn ) . Since W gives an isomorphism of prefactorization algebras between Obsq and (Obscl [h], ¯ h¯ ), the correlation function for the observables W(αi ) is the cohocl mology class of α1 h¯ · · · h¯ αn in Obscl (M)[h]. ¯ The map Obs (M)[h] ¯ → R[h] ¯ sending an observable α to α(0) is an R[h]-linear cochain map inducing an ¯ isomorphism on cohomology, and it sends 1 to 1. There is a unique such map (unique up to cochain homotopy). Therefore,  W(α1 ) · · · W(αn ) = α1 h¯ · · · h¯ αn (0), as claimed. This statement does not resemble the usual statement in a physics text, so we unpack it in an example to clarify the relationship with other versions of Wick’s lemma. As an example, let us suppose that we have two linear observables α1 , α2 of cohomological degree 0, defined on open sets U, V. Thus, α1 ∈ Cc∞ (U) and

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α2 ∈ Cc∞ (V). As these are linear observables, the second-order operator ∂G does nothing, so W(αk ) = αk . But the product is quadratic so we have W −1 (α1 α2 ) = α1 α2 − h∂ ¯ G (α1 α2 ). As

 ∂G (α1 α2 ) =

G(x, y)α1 (x)α2 (y), M×M

it follows that

 W(α1 )W(α2 ) = −h¯

G(x, y)α1 (x)α2 (y), M×M

which is what a physicist would write down as the expected value of two linear observables. Remark: We have set things up so that we are computing the functional integral against the measure eS/h¯ dμ ,where S is the action functional and dμ is the (non-existent) “Lebesgue measure” on the space of fields. Physicists often use the measures e−S/h¯ or eiS/h¯ . By changing h¯ (e.g., h¯ → −h), ¯ one can move between these different conventions. ♦.

4.8 Translation-Invariant Prefactorization Algebras In this section we will analyze in detail the notion of translation-invariant prefactorization algebras on Rn . Most theories from physics possess this property. For one-dimensional field theories, one often uses the phrase “time-independent Hamiltonian” to indicate this property, and in this section we will explain, in examples, how to relate the Hamiltonian formalism of quantum mechanics to our approach.

4.8.1 The Definition We now turn to the definition of a translation-invariant prefactorization algebra. It is a special case of the definition of an equivariant prefactorization algebra in Section 3.7 in Chapter 3, but we go through the definition here in some detail. Remark: We use the term “translation-invariance” here in place of translationequivariance. The motivation is that a field theory on Rn whose action functional is translation-invariant will produce a translation-equivariant prefactorization algebra. We hope this slight inconsistency of terminology causes no confusion. ♦

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If U ⊂ Rn and x ∈ Rn , let τx U = {y : y − x ∈ U} denote the translate of U by x. Definition 4.8.1 A prefactorization algebra F on Rn is discretely translationinvariant if we have isomorphisms τx : F(U) ∼ = F(τx U) for all x ∈ Rn and all open subsets U ⊂ Rn . These isomorphisms must satisfy a few conditions. First, we require that τx ◦ τy = τx+y for every x, y ∈ Rn . Second, for all disjoint open subsets U1 , . . . , Uk in V, the diagram F(U1 ) ⊗ · · · ⊗ F(Uk )  F(V)

τx

τx

/ F(τx U1 ) ⊗ · · · ⊗ F(τx Uk )  / F(τx V)

commutes. (Here the vertical arrows are the structure maps of the prefactorization algebra.) Example: Consider the prefactorization algebra of quantum observables of the free scalar field theory on Rn , as defined in Section 4.2.6. This theory has as its complex of fields   ∞  ∞ → C [−1] . E = C − By definition, Obsq (U) is the Chevalley–Eilenberg chains of a −1-shifted central extension Ec (U) of Ec (U), with cocycle defined by φ 0 φ 1 where φ k denotes the field in cohomological degree k. This Heisenberg algebra is defined using only the Riemannian structure on Rn , and it is therefore automatically invariant under all isometries of Rn . In particular, the resulting prefactorization algebra is discretely translationinvariant. ♦ We are interested in a refined version of this notion, where the structure maps of the prefactorization algebra depend smoothly on the position of the open sets. It is a bit subtle to talk about “smoothly varying an open set,” and to do this, we introduce some notation. Recall the notion of a derivation of a prefactorization algebra on a manifold M.

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Definition 4.8.2 A degree k derivation of a prefactorization algebra F is a collection of maps DU : F(U) → F(U) of cohomological degree k for each open subset U ⊂ M, with the property that for any finite collection U1 , . . . , Un ⊂ V of disjoint opens and an element αi ∈ F(Ui ) for each open, the derivation acts by a Leibniz rule on the structure maps:  1 ,...,Un (α1 , . . . , αn ) = (−1)k(|α1 |+···+|αi−1 |) DV mU V i U1 ,...,Un mV (α1 , . . . , DUi αi , . . . , αn ),

where the sign arises from the usual Koszul sign rule. Example: Let us consider, again, the prefactorization algebra of the free scalar field on Rn . Observables on U are Chevalley–Eilenberg chains of the Heisenberg algebra Ec (U). Let X denote a Killing vector field on Rn (i.e., X is an infinitesimal isometry). For example, we could take X to be a translation vector field ∂x∂ j . Then the Heisenberg dg Lie algebra has a derivation where φ k → Xφ k , for k = 0, 1, and the central element h¯ is annihilated. (Recall that φ k is notation for an element of the copy of Cc∞ (U) situated in degree k.) Because X is a Killing vector field on Rn , it commutes with the differential on the Heisenberg algebra and satisfies the equality   (Xφ 0 )φ 1 + φ 0 (Xφ 1 ) = 0. Hence, X is a derivation of dg Lie algebras. By naturality, X extends to an endomorphism of Obsq (U) = C∗ (Ec (U)). This endomorphism defines a derivation of the prefactorization algebra Obsq of observables of the free scalar field theory. ♦ All the derivations together Der∗ (F) form a differential graded Lie algebra. The differential is defined by (dD)U = [dU , DU ], where dU is the differential on F(U). The Lie bracket is defined by [D, D ]U = [DU , DU ]. The concept of derivation allows us to talk about the action of a dg Lie algebra g on a prefactorization algebra F. Such an action is simply a homomorphism of differential graded Lie algebras g → Der∗ (F). Next, we introduce some notation that will help us describe the smoothness conditions for a discretely translation-invariant prefactorization algebra. Let U1 , . . . , Uk ⊂ V be disjoint open subsets. Let Disj(U1 , . . . , Uk | V) ⊂ (Rn )k be the set of k-tuples (x1 , . . . , xk ) such that the translates τx1 U1 , . . . , τxk Uk are still disjoint and contained in V. It parametrizes the way we can move the open sets without causing overlaps. Let us assume that Disj(U1 , . . . , Uk | V)

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has nonempty interior, which happens when the closure of the Ui are disjoint and contained in V. Let F be any discretely translation-invariant prefactorization algebra. For each (x1 , . . . , xk ) ∈ Disj(U1 , . . . , Uk | V), we have a multilinear map obtained as a composition mx1 ,...,xk : F(U1 ) × · · · × F(Uk ) → F(τx1 U1 ) × · · · × F(τxk Uk ) → F(V), where the second map arises from the inclusion τx1 U1  · · ·  τxk Uk → V. Definition 4.8.3 A discretely translation-invariant prefactorization algebra F is smoothly translation-invariant if the following conditions hold. (i) The map mx1 ,...,xk above depends smoothly on (x1 , . . . , xk ) ∈ Disj (U1 , . . . , Uk | V). (ii) The prefactorization algebra F is equipped with an action of the Abelian Lie algebra Rn of translations. If v ∈ Rn , we will denote the corresponding action maps by d : F(U) → F(U). dv We view this Lie algebra action as an infinitesimal version of the global translation invariance. (iii) The infinitesimal action is compatible with the global translation invariance in the following sense. For v ∈ Rn , let vi ∈ (Rn )k denote the vector (0, . . . , v, . . . , 0), with v placed in the i-slot and 0 in the other k − 1 slots. If αi ∈ F(Ui ), then we require that

 d d mx ,...,x (α1 , . . . , αk ) = mx1 ,...,xk α1 , . . . , αi , . . . , αk . dvi 1 k dv When we refer to a translation-invariant prefactorization algebra without further qualification, we will always mean a smoothly translation-invariant prefactorization algebra. Example: We have already seen that the prefactorization algebra of the free scalar field theory on Rn is discretely translation invariant and is also equipped with an action of the Abelian Lie algebra Rn by derivations. It is easy to verify that this prefactorization algebra is smoothly translation-invariant. ♦ Example: This example is a special case of the example in Section 3.7.3. Let F be the cohomology of the prefactorization algebra of observables of the free scalar field theory on R with mass m. We have seen in Section 4.3 that the corresponding associative algebra A is the Weyl algebra, generated by p, q, h¯ with commutation relation [p, q] = h. ¯

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It is straightforward to verify that F is a locally constant, smoothly translation invariant prefactorization algebra on R, valued in vector spaces. As the action of R on A is smooth, it differentiates to an infinitesimal action of the ∂ Lie algebra R on A by derivations. The basis element ∂x of R becomes a derivation H of A, called the Hamiltonian. The Hamiltonian here is given by ♦ H(a) = 21h¯ [p2 − m2 q2 , a].

4.8.2 An Operadic Reformulation Next, we will explain how to think of the structure of a translation-invariant prefactorization algebra on Rn in more operadic terms. This description has a lot in common with the En algebras familiar from topology. Let r1 , . . . , rk , s ∈ R>0 . Let Discsn (r1 , . . . , rk | s) ⊂ (Rn )k be the (possibly empty) open subset consisting of k-tuples x1 , . . . , xk ∈ Rn with the property that the closures of the balls Bri (xi ) are all disjoint and contained in Bs (0). Here, Br (x) denotes the open ball of radius r around x. Definition 4.8.4 Let Discsn be the R>0 -colored operad in the category of smooth manifolds whose space of k-ary morphisms Discsn (r1 , . . . , rk | s) between ri , s ∈ R>0 is described previously. Note that a colored operad is the same thing as a multicategory. An R>0 colored operad is thus a multicategory whose set of objects is R>0 . The essential data of the colored operad structure on Discsn are the following. We have maps ◦i : Discsn (r1 , . . . , rk | ti )× Discsn (t1 , . . . , tm | s) → Discsn (t1 , . . . , ti−1 , r1 , . . . , rk , ti+1 , . . . , tm | s). This map is defined by inserting the outgoing ball (of radius ti ) of a configuration x ∈ Discsn (r1 , . . . , rk | ti ) into the ith incoming ball of a point y ∈ Discsn (t1 , . . . , tk | s). These maps satisfy the natural associativity and commutativity properties of a multicategory. Next, let F be a translation-invariant prefactorization algebra on Rn . Let Fr = F(Br (0)) denote the cochain complex that F assigns to a ball of radius r. This notation is reasonable because translation invariance gives us an isomorphism between F(Br (0)) and F(Br (x)) for any x ∈ Rn .

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The structure maps for a translation-invariant prefactorization algebra yield, for each configuration p ∈ Discsn (r1 , . . . , rk | s), a multiplication operation m[p] : Fr1 × · · · × Frk → Fs . The map m[p] is a smooth multilinear map of differentiable spaces; and furthermore, this map depends smoothly on p. These operations make the complexes Fr into an algebra over the R>0 colored operad Discsn (r1 , . . . , rk | s), valued in the multicategory of differentiable cochain complexes. In addition, the complexes Fr are endowed with an action of the Abelian Lie algebra Rn . This action is by derivations of the Discsn -algebra F compatible with the action of translation on Discsn , as described previously. Now, let us unravel explicitly what it means to be such a Discsn algebra. The first property is that, for each configuration p ∈ Discsn (r1 , . . . , rk | s), the map m[p] is a multilinear map of cohomological degree 0, compatible with differentials. Second, let N be a manifold and let fi : N → Frdii be smooth maps into the space Frdii of elements of degree di . The smoothness properties of the map m[p] mean that the map N × Discsn (r1 , . . . , rk | s) → (x, p) →

Fs m[p](f1 (x), . . . , fk (x))

is smooth. Thus, we can work with smooth families of multiplications. Next, note that a permutation σ ∈ Sk gives an isomorphism σ : Discsn (r1 , . . . , rk | s) → Discsn (rσ (1) , . . . , rσ (k) | s). We require that for each p ∈ Discsn (r1 , . . . , rk | s) and collection of elements αi ∈ Fri , m[σ (p)](ασ (1) , . . . , ασ (k) ) = m[p](α1 , . . . , αk ). Finally, we require that the maps m[p] are compatible with composition, in the following sense. For any configurations p ∈ Discsn (r1 , . . . , rk | ti ) and q ∈ Discsn (t1 , . . . , tl | s) and for collections αi ∈ Fri and βj ∈ Ftj , we require that m[q](β1 , . . . , βi−1 , m[p](α1 , . . . , αk ), βi+1 , . . . , βl ) = m[q ◦i p](β1 , . . . , βi−1 , α1 , . . . , αk , βi+1 , . . . , βl ). In addition, the action of Rn on each F r is compatible with these multiplication maps, in the manner described in the preceding text.

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137

4.8.3 A Cooperadic Reformulation Let us give one more equivalent way of rewriting these axioms, which will be useful when we discuss the holomorphic context. These alternative axioms will say that the spaces C∞ (Discsn (r1 , . . . , rk | s)) form an R>0 -colored co-operad when we use the appropriate completed tensor product. Since we know how to tensor a differentiable vector space with the space of smooth functions on a manifold, it makes sense to talk about an algebra over this colored cooperad in the category of differentiable cochain complexes. We can rephrase the smoothness axiom for the product map m[p] : Fr1 ⊗ · · · ⊗ Frk → Fs , as p varies in Discsn (r1 , . . . , rk | s), as follows. For any differentiable vector space V and smooth manifold M, we use the notation V ⊗C∞ (M) interchangeably with the notation C∞ (M, V); both indicate the differentiable vector space of smooth maps M → V. The smoothness axiom states that the map above extends to a smooth map of differentiable spaces μ(r1 , . . . , rk | s) : Fr1 × · · · × Frk → Fs ⊗ C∞ (Discsn (r1 , . . . , rk | s)). In general, if V1 , . . . , Vk , W are differentiable vector spaces and if X is a smooth manifold, let C∞ (X, HomDVS (V1 , . . . , Vk | W)) denote the space of smooth multilinear maps V1 × · · · × Vk → C∞ (X, W). Note that there is a natural gluing map ◦i : C∞ (X, HomDVS (V1 , . . . , Vk | Wi )) × C∞ (Y, HomDVS (W1 , . . . , Wl | T)) → C∞ (X × Y, HomDVS (W1 , . . . , Wi−1 , V1 , . . . , Vk , Wi+1 , . . . , Wl | T)). With this notation in hand, there are elements  μ(r1 , . . . , rk | s) ∈ C∞ Discs(r1 , . . . rk | s), HomDVS (Fr1 , . . . , Frk | Fs ) with the following properties. (i) The map μ(r1 , . . . , rk | s) is closed under the natural differential, arising from the differentials on the cochain complexes Fri . (ii) If σ ∈ Sk , then σ∗ μ(r1 , . . . , rk | s) = μ(rσ (1) , . . . , rσ (k) | s)

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where σ∗ : C∞ ( Discs(r1 , . . . rk | s), HomDVS (Fr1 , . . . , Frk | Fs )) → C∞ (Discs(rσ (1) , . . . rσ (k) | s), HomDVS (Frσ (1) , . . . , Frσ (k) | Fs )) is the natural isomorphism. (iii) As before, let ◦i : Discsn (r1 , . . . , rk | ti ) × Discsn (t1 , . . . , tm | s) → Discsn (t1 , . . . , ti−1 , r1 , . . . , rk , ti+1 , . . . , tm | s) denote the gluing map. Then we require that ◦∗i μ(t1 , . . . , ti−1 , r1 , . . . , rk , ti+1 , . . . , tl ) = μ(r1 , . . . , rk | ti ) ◦i μ (t1 , . . . , tl | s). These elements equip the Fr with the structure of an algebra over the colored cooperad, as stated earlier.

4.8.4 The Free Scalar Field Let us write down explicit formulas for these product maps, as a Discsn algebra, in the case of the free scalar field theory. We have seen in Section 4.6 that the choice of a Green’s function G leads to an isomorphism of cochain q complexes WU : Obscl (U)[h] ¯ → Obs (U)[h] ¯ for every open subset U. This isomorphism allows us to transfer the product (or structure map) in the prefactorization algebra Obsq to a deformed product h¯ in the prefactorization algebra Obscl [h], ¯ defined by −1 (WU (α) · WV (β)), α h¯ β = WUV cl where U and V are disjoint open sets, α ∈ Obscl (U)[h], ¯ β ∈ Obs (V)[h], ¯ and the dot · indicates the structure map in the prefactorization algebra Obsq . This isomorphism leads to a completely explicit description of the product maps

μsr1 ,...,rk : Fr1 ⊗ · · · ⊗ Frk → C∞ (P(r1 , . . . , rk | s), Fs ) discussed earlier, in the case that F arises from the prefactorization algebra of quantum observables of a free scalar field theory, or equivalently from the prefactorization algebra (Obscl [h], ¯ h¯ ). For example, on R2 , let α1 , α2 be compactly supported smooth functions on discs Bri (0) of radii ri around 0. Let us view each αi as a cohomological degree

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139

0 element of ∞ ∞ Fri = Obscl (Bri (0))[h] ¯ = Sym(Cc (Bri (0))[1] ⊕ Cc (Bri (0)))[h]. ¯

Let τx αi denote the translate of αi to an element of Cc∞ (Bri (x)). Then, for x1 , x2 ∈ R2 such that the Bri (xi ) are disjoint and contained in Bs (0), we have μsr1 ,r2 (α1 , α2 ) = τx1 α1 h¯ τx2 α2 = (τx1 α1 ) · (τx2 α2 ) − h¯  × α1 (u1 + x1 )α2 (u2 + x2 ) u1 ,u2 ∈R2

× log |u1 − u2 | . 1 Here, we used the Green’s function G(x, y) = − 2π log |x − y|.

4.9 States and Vacua for Translation Invariant Theories We develop notions such as state and vacuum in this setting of translationinvariant prefactorization algebras. Our running example is the free scalar field. As should be clear from their construction, the prefactorization algebra of observables only encodes the local relationships between the observables. Longrange and global aspects of a physical situation appear in a different way, and this notion of state provides one method by which to introduce them.The examples here exhibit the role of fixing constraints on the asymptotic or boundary behavior of the fields. Remark: We do not pursue here a comparison with classic axioms for quantum field theory, such as the Wightman or Osterwalder–Schrader axioms, but the familiar reader will recognize the relationship. Both Kazhdan’s lectures in Deligne et al. (1999) and Glimm and Jaffe (1987) cover such material and much more. ♦ Definition 4.9.1 Let F be a smoothly translation-invariant prefactorization algebra on Rn , over a ring R. (In practice, R is R, C or R[[h]], ¯ C[[h]]). ¯ ∗ n A state for F is a smooth linear map − : H (F(R )) → R. A state − is translation invariant if it commutes with the action of both the group Rn and of the infinitesimal action of the Lie algebra Rn , where Rn acts trivially on R.

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Recall that “smooth” means the following. The ring R itself is a differentiable vector space: for instance, the real numbers R as a topological vector space corresponds to the sheaf C∞ on the site of smooth manifolds. As F(Rn ) is a differentiable vector space, it is also a sheaf on the site of smooth manifolds, and H ∗ (F(Rn )) denotes the cohomology sheaf on the site of smooth manifolds. Unraveling the structures in the preceding definition, we find that for each smooth manifold M and each element α ∈ H ∗ (C∞ (M, F(Rn )), the element α is a smooth R-valued function on M. A state − allows us to define correlation functions of observables (even if the complex of fields is not acyclic). If Oi ∈ F(Bri (0)) are observables, then we can construct τxi Oi ∈ F(Bri (xi )). If the configuration (x1 , . . . , xk ) satisfies the condition that the discs Bri (xi ) are disjoint – that is, (x1 , . . . , xk ) ∈ P(r1 , . . . , rk | ∞) – then we define the correlation function O1 (x1 ) · · · On (xn ) by applying the state − to the product observable τx1 O1 · · · τxn On ∈ F(Rn ). Because we have a smoothly translation invariant prefactorization algebra and the state is a map of differentiable cochain complexes, we see that O1 (x1 ) · · · On (xn ) is a smooth function of the points xi , i.e., an element of C∞ (P(r1 , . . . , rk | ∞), R). Further, this function is invariant under simultaneous translation of all the points xi . Definition 4.9.2 A translation-invariant state − is a vacuum if it satisfies the cluster decomposition principle: in the preceding situation, for any two observables O1 ∈ F(Br1 (0)) and O2 ∈ F(Br2 (0)), we have O1 (0) O2 (x) − O1 (0) O2 (0) → 0 as x goes to infinity. A vacuum is massive if O1 (0) O2 (x) − O1 (0) O2 (0) tends to zero exponentially fast. Example: This extended example examines vacua of the free scalar field with mass m. We have seen that the choice of a Green’s function G for the operator +m2 leads to an isomorphism of cochain complexes Obscl (U)[h] = Obsq (U) for ¯ ∼ n every open subsets U ⊂ R . This produces an isomorphism of prefactorization algebras if we endow Obscl [h] ¯ with a deformed factorization product h¯ , defined using the Green’s function. If m > 0, there is a unique Green’s function G of the form G(x, y) = f (x−y), where f is a distribution on Rn that is smooth away from the origin and tends to zero exponentially fast at infinity. For example, if n = 1, the function f is 1 −m|x| e . f (x) = 2m

4.9 States and Vacua for Translation Invariant Theories If m = 0, we already know $ G(x, y) =

1 (d/2 − 1) |x − y|2−n 4π d/2 1 − 2π log |x − y|

141

if n = 2 if n = 2

is the canonically defined Green’s function. It is standard to interpret the Green’s function as the two-point correlation function δx δ0 . Hence, we see that this two-point function exhibits the necessary behavior: for m > 0, it decays exponentially fast, and it decays slowly in the massless case. Strictly speaking, this observable is not in our prefactorization algebra, as it is distributional, but it does exist at the cohomological level. Alternatively, we could work with smeared versions. Because Obscl (Rn ) is, as a cochain complex, the symmetric algebra on the +m2

complex Cc∞ (Rn )[1] −−−→ Cc∞ (Rn ), there is a map from Obscl (Rn ) to R that is the identity on Sym0 and sends Sym>0 to 0. (In other words, we have a natural augmentation map.) This map extends to an R[h]-linear cochain map ¯ − : Obscl (Rn )[h] ¯ → R[h]. ¯ Clearly, this map is translation invariant and smooth. Thus, because we have a q cochain isomorphism between Obscl (Rn )[h] ¯ and Obs (Rn ), we have produced a translation-invariant state. Lemma 4.9.3 For m > 0, this state is a massive vacuum. For m = 0, this state is a vacuum if n > 2; for n ≤ 2, it does not satisfy the cluster decomposition principle. Proof Let F1 ∈ Cc∞ (D(0, r1 ))⊗k1 and F2 ∈ Cc∞ (D(0, r2 ))⊗k2 . We will view F1 , F2 as observables in Obscl (Bri (0)) by using the natural map from Cc∞ (U)⊗k to the coinvariants Symk Cc∞ (U). Let τc F2 be the translation of F2 by c, where c is sufficiently large so that the discs D(0, r1 ) and D(c, r2 ) are disjoint. Explicitly, τc F2 is represented by the function F2 (y1 − c, . . . , yk2 − c). We are interested in computing the expected value F1 (τc F2 ) . In the preceding text we gave an explicit formula for the quantum factorization product h¯ on Obscl [h]. ¯ In this case, it states that F1 h¯ τc F2 is given by min(k 1 ,k2 )  r=0

h¯ r

 1≤i1 1, the structure we find is a higher-dimensional analog of a vertex algebra. (We leave a detailed analysis of the structure present in the higher-dimensional case for future work). Such higher-dimensional vertex algebras appear, for example, as the prefactorization algebra of observables of partial twists of supersymmetric gauge theories.

5.1 Vertex Algebras and Holomorphic Prefactorization Algebras on C In mathematics, the notion of a vertex algebra is a standard formalization of the observables of a two-dimensional chiral field theory. In the version of the axioms we consider, we do not ask for the Virasoro algebra to act. Before embarking on our own approach, we recall the definition of a vertex algebra and various properties as given in Frenkel and Ben-Zvi (2004).  Definition 5.1.1 Let V be a vector space. An element a(z) = n∈Z an z−n in End V[[z, z−1 ]] is a field if, for each v ∈ V, there is some N such that aj v = 0 for all j > N. Remark: The usage of the term “field” in the theory of vertex operators often provokes confusion. In this book, the term field is used to refer to a configuration in a classical field theory: for example, in a scalar field theory on a 145

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manifold M, a field is an element of C∞ (M). The term “field” as used in the theory of vertex algebras is not an example of this usage. Instead, as we will see in the text that follows, a “field” in a vertex algebra corresponds to an observable with support at a point of the surface; it can be understand as a special kind of operator. ♦ Definition 5.1.2 [Definition 1.3.1 in, Frenkel and Ben-Zvi (2004)] A vertex algebra is the following data: • • • •

A vector space V over C (the state space) A nonzero vector |0 ∈ V (the vacuum vector) A linear map T : V → V (the shift operator) A linear map Y(−, z) : V → End V[[z, z−1 ]] sending every vector v to a field (the vertex operation)

subject to the following axioms: • (Vacuum axiom) Y(|0 , z) = idV and Y(v, z)|0 ∈ v + zV[[z]] for all v ∈ V • (Translation axiom) [T, Y(v, z)] = ∂z Y(v, z) for every v ∈ V and T|0 = 0 • (Locality axiom) For any pair of vectors v, v ∈ V, there exists a nonnegative integer N such that (z − w)N [Y(v, z), Y(v , w)] = 0 as an element of End V[[z±1 , w±1 ]]. The vertex operation is best understood in terms of the following intuition. The vector space V represents the set of pointwise measurements one can make of the fields, and one should imagine labeling each point z ∈ C by a copy of V, which we’ll denote Vz . Moreover, in a disc D containing the point z, the measurements at z are a dense subspace of the measurements one can make in D (this is a feature of chiral theories). We’ll denote the observables on D by VD . The vertex operation is a way of combining pointwise measurements. Let D be a disc centered on the origin. For z = 0, we can multiply observables to get a map

z D

0



Yz : V0 ⊗ Vz → VD .

5.1 Vertex Algebras and Holomorphic Prefactorization Algebras on C147 This vertex operator should vary holomorphically in z ∈ D \ {0}. In other words, we should get something with properties like the formal definition Y(−, z) above. (This picture clearly resembles the “pair of pants” product from two-dimensional topological field theories, although we’ve shrunk the two “incoming” boundary circles to points.) An appealing aspect of our approach to observables is that this intuition becomes explicit and rigorous. Our procedure provides structure maps from disjoint discs into a larger disc (and also describes how to “multiply” observables supported in more complicated opens). Moreover, it gives the structure maps in a coordinate-free way. By choosing a coordinate z on C, we recover the usual formulas for vertex algebras. This relationship is seen in the main theorem in this chapter, which we now state. The theorem connects vertex algebras with a certain class of prefactorization algebras on C. The prefactorization algebras of interest are holomorphically translation invariant prefactorization algebra. We will give a precise definition of what this means shortly, but here is a rough definition. Let F be such a prefactorization algebra. Let Fr denote the cochain complex F(D(0, r)) that F assigns to a disc of radius r. Recall, as we explained in Section 4.8 of Chapter 4, if F is smoothly translation invariant, then we have an operator product map μ : Fr1 ⊗ · · · ⊗ Frn → C∞ (Discs(r1 , . . . , rk | s), Fs ), where Discs(r1 , . . . , rk | s) refers to the open subset of Ck consisting of points z1 , . . . , zk such that the discs of radius ri around zi are all disjoint and contained in the disc of radius s around the origin. If F is holomorphically translation invariant, then μ lifts to a cochain map μ : Fr1 ⊗ · · · ⊗ Frn → 0,∗ (Discs(r1 , . . . , rk | s), Fs ), where on the right-hand side we have used the Dolbeault complex of the complex manifold Discs(r1 , . . . , rk | s). We also require some compatibility of these lifts with compositions, which we will detail later. In other words, being holomorphically translation invariant means that the operator product map is holomorphic (up to homotopy) in the location of the discs. Theorem 5.1.3 Let F be a holomorphically translation invariant prefactorization algebra on C. Let F be equivariant under the action of S1 on C by rotation, and let Frk denote the weight k eigenspace of the S1 action on the complex Fr . Assume that for every r < s, the extension map Frk → Fsk associated to the inclusion D(0, r) ⊂ D(0, s) is a quasi-isomorphism. Finally, we need to

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assume that the S1 action on each Fr satisfies a certain technical “tameness” condition. Then the vector space  VF = H ∗ (Frk ) k∈Z

has the structure of a vertex algebra. The vertex algebra structure map YF : VF ⊗ VF → VF [[z, z−1 ]] is the Laurent expansion of operator product map H ∗ μ : H ∗ (Frk11 ) ⊗ H ∗ (Frk22 ) → Hol(Discs(r1 , r2 | s), H ∗ (Fs )). On the right-hand side, Hol denotes the space of holomorphic maps. In other words, the intuition that the vertex algebra structure map is the operator product expansion is made precise in our formalism. This result should be compared with the classic result of Huang (1997), who relates vertex algebras with chiral conformal field theories at genus 0, in the sense used by Segal. As we have seen in Section 1.4 in Chapter 1, the axioms for prefactorization algebras are closely related to Segal’s axioms. Our axioms for a holomorphically translation invariant field theory are similarly related to Segal’s axioms for a two-dimensional chiral field theory. Although our result is closely related to Huang’s, it is a little different because of the technical differences between a prefactorization algebra and a Segal-style chiral conformal field theory. One nice feature of our definition of holomorphically translation-invariant prefactorization algebra is that it makes sense in any complex dimension. The structure present on the cohomology of a higher-dimensional holomorphically translation-invariant prefactorization algebra is a higher-dimensional version of the axioms of a vertex algebra. This structure is discussed briefly in Costello and Scheimbauer (2015). Another nice feature of our approach is that our general construction of field theories allows one to construct vertex algebras by perturbation theory, starting with a Lagrangian. This method should lead to the construction of many interesting vertex algebras, and of their higher dimensional analogs.

5.1.1 Organization of this Chapter We start this chapter by stating and proving our main theorem. The first thing we define, in Section 5.2, is the notion of a holomorphically translation invariant prefactorization algebra on Cn for every n ≥ 1. Holomorphic translation

5.2 Holomorphically Translation-Invariant Prefactorization Algebras 149

invariance guarantees that the operator products are all holomorphic. We will then show to construct a vertex algebra from such an object in dimension 1. The rest of this chapter is devoted to analyzing examples. In Section 5.4, we discuss the prefactorization algebra associated to a very simple two-dimensional chiral conformal field theory: the free βγ system. We will show that the vertex algebra associated to the prefactorization algebra of observables of this theory is an object called the βγ vertex algebra in the literature. Then, in Section 5.5, we will construct a prefactorization algebra encoding the affine Kac–Moody algebra. This prefactorization algebra again encodes a vertex algebra, which is the standard Kac–Moody vertex algebra.

5.2 Holomorphically Translation-Invariant Prefactorization Algebras In this section we analyze in detail the notion of translation-invariant prefactorization algebras on Cn . On Cn we can ask for a translation-invariant prefactorization algebra to have a holomorphic structure; this implies that all structure maps of the prefactorization algebra are (in a sense we will explain shortly) holomorphic. There are many natural field theories where the corresponding prefactorization algebra is holomorphic: for instance, chiral conformal field theories in complex dimension 1, and minimal twists of supersymmetric field theories in complex dimension 2, as described in Costello (2011b).

5.2.1 The Definition We now explain what it means for a (smoothly) translation-invariant prefactorization algebra F on Cn to be holomorphically translation-invariant. For this definition to make sense, we require that F is defined over C: that is, the vector spaces F(U) are complex vector spaces and all structure maps are complexlinear. Recall that such a prefactorization algebra has, as part of its structure, an action of the real Lie algebra R2n = Cn by derivations. This action is as a real Lie algebra. Since F is defined over C, the action extends to an action of the complexified translation Lie algebra R2n ⊗R C. We will denote the action maps by ∂ ∂ , : F(U) → F(U). ∂zi ∂zj Definition 5.2.1 A translation-invariant prefactorization algebra F on Cn is holomorphically translation-invariant if it is equipped with derivations ηi :

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F → F of cohomological degree −1, for i = 1 . . . n, with the following properties: ∂ ∈ Der(F) ∂zi [ηi , ηj ] = 0, ' ( ∂ ηi , = 0. ∂zi dηi =

Here, d refers to the differential on the dg Lie algebra Der(F). We should understand this definition as saying that the vector fields ∂z∂ i act homotopically trivially on F. In physics terminology, the operators ∂x∂ i , where xi are real coordinates on Cn = R2n , are related to the energy-momentum tensor. We are asking that the components of the energy-momentum tensor in the zi directions are exact for the differential on observables (in physics, this might be called “BRST exact”). This is rather similar to a phenomenon that sometimes appears in the study of topological field theory, in which a topological theory depends on a metric, but the variation of the metric is exact for the BRST differential. See Witten (1988b) for discussion.

5.2.2 An Operadic Reformulation Now we will interpret holomorphically translation-invariant prefactorization algebras in the language of R>0 -colored operads. When we work in complex geometry, it is better to use polydiscs instead of balls, as is standard in complex analysis. Thus, if z ∈ Cn , let PDr (z) = {w ∈ Cn | |wi − zi | < r for 1 ≤ i ≤ n} denote the polydisc of radius r around z. Let PDiscsn (r1 , . . . , rk | s) ⊂ (Cn )k denote the set of z1 , . . . , zk ∈ Cn with the property that the closures of the polydiscs PDri (zi ) are disjoint and contained in the polydisc PDs (0). The spaces PDiscsn (r1 , . . . , rk | s) form a R>0 -colored operad in the category of complex manifolds. We will explain here why a holomorphically translation invariant prefactorization algebra provides an algebra over this colored operad. Proposition 5.2.2 will provide a precise version and proof of this argument.

5.2 Holomorphically Translation-Invariant Prefactorization Algebras 151 Now, let F be a holomorphically translation-invariant prefactorization algebra on Cn . Let Fr denote the differentiable cochain complex F(PDr (0)) associated to the polydisc of radius r. For each p ∈ PDiscsn (r1 , . . . , rk | s), we have a map m[p] : Fr1 × · · · × Frk → Fs . This map is smooth, multilinear, and compatible with the differential. Further, this map varies smoothly with p. The fact that F is a holomorphically translation-invariant prefactorization algebra means that these maps are equipped with extra structure: we have derivations ηj of F that make the derivations ∂z∂ j homotopically trivial. For i = 1, . . . , k and j = 1, . . . , n, let zij , zij refer to coordinates on (Cn )k . We thereby obtain coordinates on the open subset PDiscsn (r1 , . . . , rk | s) ⊂ (Cn )k . Thus, we have operations ∂ m[p] : Fr1 × · · · × Frk → Fs ∂zij obtained by differentiating the operation m[p], which depends smoothly on p, in the direction zij . Let m[p] ◦i ηj denote the operation m[p] ◦i ηj : Fr1 × · · · × Frk → Fs α1 × · · · × αk → (−1)|α1 |+···|αi−1 | m[p](α1 , . . . , ηj αi , . . . , αk ), where the sign arises from the usual Koszul rule. The axioms of a (smoothly) translation invariant prefactorization algebra tell us that ∂ ∂ m[p] = m[p] ◦i , ∂zij ∂zj where

∂ ∂zj

is the derivation of the prefactorization algebra F. This equality,

together with the fact that [d, ηi ] =

∂ ∂zj ,

tells us that

* ) ∂ m[p] = d, m[p] ◦i ηj ∂zij holds. Hence the product map m[p] is holomorphic in p, up to a homotopy given by ηi .

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5.2.3 A Cooperadic Reformulation In the smooth case, we saw that we could also describe the structure as that of an algebra over a R>0 -colored cooperad built from smooth functions on the spaces Discsn (r1 , . . . , rk | s). In this section we will see that there is an analogous story in the complex world, where we use the Dolbeault complex of the spaces PDiscsn (r1 , . . . , rk | s). Because the spaces PDiscsn (r1 , . . . , rk | s) form a colored operad in the category of complex manifolds, their Dolbeault complexes form a colored cooperad in the category of convenient cochain complexes. We use here that the contravariant functor sending a complex manifold to its Dolbeault complex, viewed as a convenient vector space, is symmetric monoidal: β 0,∗ (Y) 0,∗ (X × Y) = 0,∗ (X)⊗ β denotes the symmetric monoidal structure on the category of convewhere ⊗ nient vector spaces. Explicitly, the colored cooperad structure is given as follows. The operad structure on the complex manifolds PDiscsn (r1 , . . . , rk | ti ) is given by maps ◦i : PDiscsn (r1 , . . . , rk | ti ) × PDiscsn (t1 , . . . , tm | s) → PDiscsn (t1 , . . . , ti−1 , r1 , . . . , rk , ti+1 , . . . , tm | s). We let ◦∗i : 0,∗ (PDiscsn (t1 , . . . , ti−1 , r1 , . . . , rk , ti+1 , . . . , tm | s)) → 0,∗ (PDiscsn (r1 , . . . , rk | ti ) × PDiscsn (t1 , . . . , tm | s)) be the corresponding pullback map on Dolbeault complexes. The prefactorization algebras we are interested in take values in the category of differentiable vector spaces. We want to say that if F is a holomorphically translation invariant prefactorization algebra on Cn , then it defines an algebra over the colored cooperad given by the Dolbeault complex of the spaces PDiscsn . A priori, this does not make sense, because the category of differentiable vector spaces is not a symmetric monoidal category; only its full subcategory of convenient vector spaces has a tensor structure. However, to make this definition, all we need to be able to do is to tensor differentiable vector spaces with the Dolbeault complexes of complex manifolds, and this we know how to do. As shown in Appendix B, for any manifold X, C∞ (X) defines a differentiable vector space (in fact, a commutative algebra in DVS). If V is any differentiable vector space, then there is a differentiable vector space C∞ (X, V)

5.2 Holomorphically Translation-Invariant Prefactorization Algebras 153 whose value on a manifold M is C∞ (M × X, V). If X is a complex manifold, then 0,∗ (X) is a differentiable cochain complex, for the same reasons. We define 0,∗ (X, V) as the tensor product 0,∗ (X, V) := 0,∗ (X) ⊗C∞ (X) C∞ (X, V) as sheaves of C∞ (X)-modules on the site of smooth manifolds. This is a reasonable thing to do as 0,∗ (X) is a projective module over C∞ (X). Note also that the fact that V is a differentiable vector space, and not just a sheaf of C∞ modules on the site of smooth manifolds, means that differential operators on X act on C∞ (X, V). This is what allows us to extend the differential ∂ on the Dolbeault complex 0,∗ (X) to an operator on 0,∗ (X, V). The Dolbeault complex with coefficients in V is functorial both for maps ∗ f : 0,∗ (X) → 0,∗ (Y) that arise from holomorphic maps f : Y → X and also for arbitrary smooth maps between differentiable vector spaces. Since the cooperad structure maps in our colored dg cooperad 0,∗ (PDiscsn ) arise from maps of complex manifolds, it makes sense to ask for a coalgebra over this cooperad in the category of differentiable vector spaces. Proposition 5.2.2 Let F be a holomorphically translation-invariant prefalgebra on Cn . Then F defines an algebra over the cooperad actorization 0,∗ (PDiscsn ) n∈N in differentiable cochain complexes. More precisely, the product maps m[p] : Fr1 × · · · × Frk → Fs for p ∈ PDiscsn (r1 , . . . , rk | s) lift to multilinear maps μ∂ (r1 , . . . , rk | s) : Fr1 × · · · × Frk → 0,∗ (PDiscsn (r1 , . . . , rk | s), Fs )) that are compatible with differentials and that satisfy the properties needed to define a coalgebra over a cooperad. In other words, we have closed elements μ∂ (r1 , . . . , rk | s) ∈ 0,∗ (PDiscsn (r1 , . . . , rk | s), HomDVS (Fr1 , . . . , Frk | Fs )) satisfying the following properties. (i) The element μ∂ (r1 , . . . , rk | s) is closed under the natural differential on 0,∗ (PDiscsn (r1 , . . . , rk | s), HomDVS (Fr1 , . . . , Frk | Fs )), which incorporates the Dolbeault differential as well as the internal differentials on the complexes Fri , Fs . Explicitly, the differential   (dF + ∂)μ∂ (r1 , . . . , rk | s) (f1 , . . . , fk )

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PDt (0) PDr (0) 0

PDs (z)

 Fr × Fs

μ∂

0,∗ (PDiscs(r, s | t), Ft )

z

Figure 5.1. Configurations of discs and the operator product map.

is given by the sum  ±μ∂ (r1 , . . . , rk | s)(f1 , . . . , dF fi , . . . , fk ). (ii) Let σ ∈ Sk . As in the smooth case, σ∗ μ∂ (r1 , . . . , rk | s) = μ∂ (rσ (1) , . . . , rσ (k) | s) where σ∗ is induced by the isomorphism of complex manifolds σ : PDiscsn (r1 , . . . , rk | s) → PDiscsn (rσ (1) , . . . , rσ (k) | s). (iii) For all 1 ≤ i ≤ m, ◦∗i μ∂ (t1 , . . . , ti−1 , r1 , . . . , rk , ti+1 , . . . , tm | s) = μ∂ (t1 , . . . , tm | s) ◦i μ∂ (r1 , . . . , rk | ti ). That is, we have associativity of composition. Remark: For similar axiom systems in the context of topological field theory, the interested reader should consult, for example, Getzler (1994) and Costello (2007b). In Segal (2004) there is a related system of axioms for chiral conformal field theories. This construction is also closely related to the construction of “descendants” in the physics literature on topological field theory (see, e.g., Witten (1988b, 1992)). Here is a brief explanation of the relationship. Suppose one has a field theory that depends on a Riemannian metric, but where the variation of the original action functional with respect to the metric is exact for the BRST operator (which corresponds to the differential on observables in our language). A metric-dependent functional making the variation BRST exact can be viewed as a 1-form on the moduli space of metrics. Higher homotopies yield forms on

5.2 Holomorphically Translation-Invariant Prefactorization Algebras 155

the moduli space of metrics, or, in two dimensions, on the moduli of conformal classes of metrics (i.e., on the moduli of Riemann surfaces). In our approach, because we are working with holomorphic instead of topological theories, we find elements of the Dolbeault complex of the appropriate moduli spaces of complex manifolds. In the simple situation considered here, these moduli spaces are the spaces PDiscsn . For the factorization algebras associated to holomorphic twists of supersymmetric field theories, these elements of the Dolbeault complex of PDiscsn come from the operator product of supersymmetric descendents of BPS operators. ♦ Proof of the proposition Giving a smooth multilinear map φ : Fr1 × · · · × Frk → 0,∗ (PDiscsn (r1 , . . . , rk | s), Fs ) compatible with the differentials is equivalent to giving an element of  φ ∈ 0,∗ PDiscsn (r1 , . . . , rk | s), HomDVS (Fr1 , . . . , Frk | Fs ) that is closed with respect to the differential ∂ + dF , where dF refers to the natural differential on the differentiable cochain complex of smooth multilinear maps HomDVS (Fr1 , . . . , Frk | Fs ). We will produce the desired element  μ∂ (r1 , . . . , rk | s) ∈ 0,∗ PDiscsn (r1 , . . . , rk | s), HomDVS (Fr1 , . . . , Frk | Fs starting from the operations μ0 (r1 , . . . , rk | s) ∈ C∞ (PDiscs(r1 , . . . , rk | s), HomDVS (Fr1 , . . . , Frk | s)) provided by the hypothesis that F is a smoothly translation-invariant prefactorization algebra. First, we need to introduce some notation. Recall that  k PDiscsn (r1 , . . . , rk | s) ⊂ Cn is an open (possibly empty) subset. Thus, 0,∗ (PDiscsn (r1 , . . . , rk | s)) = 0,0 (PDiscsn (r1 , . . . , rk | s)) ⊗ C[dzij ] where the dzij are commuting variables of cohomological degree 1, with i = ∂ denote the graded derivation that 1, . . . , k and j = 1, . . . , n. We let ∂(dz ij ) removes dzij . As before, let ηj : Fr → Fr denote the derivation that cobounds the derivation ∂z∂ j . We can compose any element α ∈ 0,∗ (PDiscn (r1 , . . . , rk | s), Hom(Fr1 , . . . , Frk | s))

(†)

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with ηj acting on Fri , to get α ◦i ηj ∈ 0,∗ (PDiscn (r1 , . . . , rk | s), Hom(Fr1 , . . . , Frk | s)). We use the shorthand notations: lij (α) = α ◦i ηj , ∂ , rij (α) = α ◦i ∂zj where ∂z∂ j refers to the derivation acting on Frj . Note that [dF , lij ] = rij , where dF is the differential on the graded vector space (†) above arising from the differentials on the spaces Fri , Fs . Further, the operators lij , rij all commute with each other in the graded sense. In what follows, for concision, we will write μ∂ instead of μ∂ (r1 , . . . , rk | s), and similarly for μ0 . The cochains μ∂ are defined by μ∂ = exp(−



dzij lij )μ0 .

i,j

Here dzij lij denotes the operation of wedging with dzij after applying the operator lij . Note that these two operators graded-commute. Next, we need to verify that (∂ +dF )μ∂ = 0, where ∂ +dF is the differential on graded vector space (†) above that arises by combining the ∂ operator with the differential arising from the complexes Fri , Fs . We use the following identities: [dF ,



dzij lij ] =

i,j



dzij rij ,

i,j

rij μ0 =

∂ 0 μ , ∂zij

dF μ0 = 0. The second and third identities are part of the axioms for a smoothly translationinvariant prefactorization algebra.

5.3 A General Method for Constructing Vertex Algebras

157

These identities allows us to calculate that  (∂ + dF )μ∂ = (∂ + dF ) exp(− dzij lij )μ0      = − exp − dzij lij dzij rij μ0    + exp − dzij lij (∂ + dF )μ0      ∂ dzij = exp − dzij lij − + ∂ μ0 ∂zij = 0. Thus, μ∂ is closed. It is straightforward to verify that the elements μ∂ are compatible with composition and with the symmetric group actions.

5.3 A General Method for Constructing Vertex Algebras In this section we prove that the cohomology of a holomorphically translation invariant prefactorization algebra on C with a compatible circle action gives rise to a vertex algebra. Together with the central theorem of the second volume, which allows one to construct prefactorization algebras by obstruction theory starting from the Lagrangian of a classical field theory, this gives a general method to construct vertex algebras.

5.3.1 The Circle Action Recall from Section 3.7 in Chaper 3 the definition of a smoothly G-equivariant prefactorization algebra on a manifold M with smooth action of a Lie group G. The case of interest here is the action of the isometry group S1  C of C acting on C itself. Definition 5.3.1 A holomorphically translation-invariant prefactorization algebra F on C with a compatible S1 action is a smoothly S1 R2 -invariant prefactorization algebra F, defined over the base field of complex numbers, together with an extension of the action of the complex Lie algebra LieC (S1  R2 ) = C {∂θ , ∂z , ∂z } , where ∂θ is a basis of LieC (S1 ), to an action of the dg Lie algebra C {∂θ , ∂z , ∂z } ⊕ C{η},

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where η is of cohomological degree −1 and the differential is dη = ∂z . In this dg Lie algebra, all commutators involving η vanish except for [∂θ , η] = −η. Note that, in particular, F is a holomorphically translation invariant prefactorization algebra on C. Our prefactorization algebras take values in differentiable cochain complexes. Here we will work with such vector spaces over the complex numbers; in other words, every differentiable vector space E discussed here will assign a complex vector space E(M) to a manifold M, the restriction maps are complex-linear, and so on. By their very definition, we know how to differentiate “smooth maps to a differentiable vector space.” Thus, if M is a complex manifold and E is a differentiable cochain complex, we have, for every open U ⊂ M, a cochain map ∂ : C∞ (U, E) → 0,1 (U, E). These maps form a map of sheaves of cochain complexes on M. Taking the cohomology sheaves of E, we obtain a graded differentiable vector space H ∗ (E) and we have a map of sheaves of graded vector spaces ∂ : C∞ (U, H ∗ (E)) → 0,1 (U, H ∗ (E)). The kernel of this map defines a sheaf on M, whose sections we denote by Hol(U, H ∗ (E)). We call these the holomorphic sections of H ∗ (E) on M.

The theorem on vertex algebras will require an extra hypothesis regarding the S1 -action on the prefactorization algebra. Given any compact Lie group G, we will formulate a concept of tameness for a G-action on a differentiable vector space. We will require that the S1 -action on the spaces Fr in our factorization algebra is tame. Note that for any compact Lie group G, the space D(G) of distributions on G is an algebra under convolution. Since spaces of distributions are naturally differentiable vector spaces, and the convolution product ∗ : D(G) × D(G) → D(G) is smooth, D(G) forms an algebra in the category of differentiable vector spaces. There is a map δ : G → D(G)

5.3 A General Method for Constructing Vertex Algebras

159

sending an element g to the δ-distribution at g. It is a smooth map and a homomorphism of monoids. Definition 5.3.2 Let E be a differentiable vector space and G be a compact Lie group. A tame action of G on E is a smooth action of the algebra D(G) on E. Note that this is, in particular, an action of G on E via the smooth map of groups G → D(G)× sending g to δg . If E is a differentiable cochain complex, a tame action is an action of D(G) on E that commutes with the differential on E. Let us now specialize to the case when G = S1 , which is the case relevant for the theorem on vertex algebras. For each integer k, there is an irreducible representation ρk of S1 given by the function ρk : λ → λk . (Here we use λ to denote a complex number by viewing S1 as a subset of C. ) We can view ρk as a map of sheaves on the site of smooth manifolds from the sheaf represented by the manifold S1 to the differentiable vector space C∞ , which assigns complexvalued smooth functions to each manifold. It lifts naturally to a representation ρ k of the algebra D(S1 ), defined by ρ k (φ) = φ, ρk , where −, − indicates the pairing between distributions and functions. We will abusively denote this representation by ρk , for simplicity. If E is a differentiable vector space equipped with such a smooth action of D(S1 ), we use Ek ⊂ E to denote the subspace on which D(S1 ) acts by ρk . We call this the weight k eigenspace for the S1 -action on E. In the algebra D(S1 ), the element ρk , viewed as a distribution on S1 , is an idempotent. If we denote the action of D(S1 ) on E by ∗, then the map ρk ∗ − : E → E defines a projection from E onto Ek . We will denote this projection by πk . Remark: Of course, all this holds for a general compact Lie group where instead of these ρk , we use the characters of irreducible representations. ♦

5.3.2 The Statement of the Main Theorem Now we can state the main theorem of this section. Theorem 5.3.3 Let F be a unital S1 -equivariant holomorphically translation invariant prefactorization algebra on C valued in differentiable vector spaces. Assume that, for each disc D(0, r) around the origin, the action of S1 on F(D(0, r) is tame. Let Fk (D(0, r)) be the weight k eigenspace of the S1 action on F(D(0, r)).

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We make the following additional assumptions. (i) Assume that, for r < r , the structure map Fk (D(0, r)) → Fk (D(0, r )) is a quasi-isomorphism. (ii) For k $ 0, the vector space H ∗ (Fk (D(0, r)) is zero. (iii) For each k and r, we require that H ∗ (Fk (D(0, r)) is isomorphic, as a sheaf on the site of smooth manifolds, to a countable sequential colimit of finite-dimensional graded vector spaces. Let Vk = H ∗ (Fk (D(0, r)), and let V = k∈Z Vk . (This space is independent of r by assumption.) Then V has the structure of a vertex algebra, determined by the structure maps of F. The construction is functorial: it will be manifest that a map of factorization algebras respecting all the equivariance conditions produces a map between the associated vertex algebras. We give the proof in Section 5.3.4. We spell out the data of the vertex algebra – the vacuum vector, translation operator, vertex operation, and so on – as we construct it. First, though, we discuss issues around applying the theorem. Remark: If V is not concentrated in cohomological degree 0, then it will have the structure of a vertex algebra valued in the symmetric monoidal category of graded vector spaces. That is, the Koszul rule of signs will apear in the axioms. ♦ Remark: We will often deal with prefactorization algebras F equipped with a complete decreasing filtration F i F, so that F = lim F/F i F. In this situation, to construct the vertex algebra we need that the properties listed in the theorem hold on each graded piece Gri F. This condition implies, by a spectral sequence, that they hold on each F/F i F, allowing us to construct an inverse system of vertex algebras associated to the prefactorization algebra F/F i F. The inverse limit of this system of vertex algebras is the vertex algebra associated to F. ♦

5.3.3 Using the Theorem The conditions of the theorem are always satisfied in practice by prefactorization algebras arising from quantizing a holomorphically translation invariant classical field theory. We describe in great detail the example of the free βγ system in Section 5.4.

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There is also the problem of recognizing the vertex algebra produced by such a prefactorization algebra. Thankfully, there is a useful “reconstruction” theorem that provides simple criteria to uniquely construct a vertex algebra given “generators and relations.” We will exploit this theorem to verify that we have recovered standard vertex algebras in our examples later in this chapter. Theorem 5.3.4 (Reconstruction Theorem 4.4.1 in Frenkel and Ben-Zvi (2004)) Let V be a complex vector space equipped with a nonzero vector |0 , an endomorphism T, a countable ordered set {aα }α∈S of vectors, and fields  aα (z) = aα(n) z−n−1 ∈ End(V)[[z, z−1 ]] n∈Z

such that (i) (ii) (iii) (iv)

For all α, aα (z)|0 = aα + O(z). T|0 = 0 and [T, aα (z)] = ∂z aα (z) for all α. All fields aα (z) are mutually local. V is spanned by the vectors aα(j11 ) · · · aα(jmm ) |0 with the ji < 0.

Then, using the formula Y(aα(j11 ) · · · aα(jmm ) |0 , z) =

1 : ∂ −j1 −1 (−j1 − 1)! · · · (−jm − 1)! z

aα1 (z) · · · ∂z−jm −1 aαm (z) : to define a vertex operation, we obtain a well-defined and unique vertex algebra (V, |0 , T, Y) such that Y(aα , z) = aα (z). Here : a(z)b(w) : denotes the normally ordered product of fields, defined as : a(z)b(w) := a(z)+ b(w) + b(w)a(z)− where a(z)+ =

 n≥0

an zn and a(z)− =



an zn .

n 0. We emphasize that V need not be the limit limr→0 H ∗ (F(r)). Instead, it is an algebraically tractable vector space mapping into that limit (so to speak, the finite sums of modes). We introduce now a natural partner to V into which all the H ∗ (F(r)) map. Let  V= Vk , k

where the product is taken in the category of differentiable vector spaces. There is a canonical map  πk : H ∗ (F(r)) → V k

given by the product of all the projection maps. We emphasize that V need not be the colimit colimr→∞ H ∗ (F(r)). Instead, it is an algebraically tractable

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vector space receiving a map from that colimit. Note that V is a completion of V in the sense that we now allow arbitrary, infinite sums of the modes. The Operator Product Let us start to analyze the structure on V given by the prefactorization algebra structure on F. Our approach will be to use instead the structure as an algebra over the cooperad 0,∗ (PDiscs1 ). (Since a one-dimensional polydisc is simply a disc, we will simply write disc from hereon.) We will call the structure maps for this algebra the operator product. Recall that we use μ∂ (r1 , . . . , rk | ∞) : F(r1 )×· · ·×F(rk ) → 0,∗ (Discs(r1 , . . . , rk | ∞), F(∞)) to denote such a multilinear cochain map. We want to use this operator product to produce a map mz1 ,...,zk : V ⊗k → V for every configuration (z1 , . . . , zk ) of k ordered distinct points in C. This map will vary holomorphically over that configuration space. Note that for any complex manifold X there is a truncation map   (−)0 : H ∗ 0,∗ (X, F(∞)) → Hol(X, H ∗ (F(∞))). If α ∈ 0,∗ (X, F(∞)) is a cocycle, this map is defined by first extracting the component α 0 of α in 0,0 (X, F(∞)). The fact that α is closed means that α 0 is closed for the differential DF on F(∞) and that ∂α 0 is exact. Thus, the cohomology class [α 0 ] is a section of F(∞) on X, and ∂[α 0 ] = 0 so that [α 0 ] is holomorphic. Hence, at the level of cohomology, the operator product produces a smooth multilinear map mz1 ,...,zk : H ∗ (F(r1 )) × · · · × H ∗ (F(rk )) → Hol(Discs(r1 , . . . , rk | ∞), H ∗ (F(∞))). Here zi indicate the positions of the centers of the discs in Discs(r1 , . . . , rk | ∞). Consider what happens if we shrink the discs. If ri < ri , then Discs(r1 , . . . , rk | ∞) ⊂ Discs(r1 , . . . , rk | ∞).

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We also have the extension map F(r ) → F(r), from the inclusion D(0, r ) → D(0, r). Note that the following diagram commutes: H ∗ (F(r1 )) ⊗ · · · ⊗ H ∗ (F(rk ))

/ Hol(Discs(r , . . . , r | ∞), H ∗ (F(∞))) 1 k

 H ∗ (F(r1 )) ⊗ · · · ⊗ H ∗ (F(rk ))

 / Hol(Discs(r1 , . . . , rk | ∞), H ∗ (F(∞))).

Because V maps to limr→0 H ∗ (F(r)), the operator product, when restricted to V, gives a map ∗

(F (∞)) mzH1 ,...,z : V ⊗k → lim Hol(Discs(r, . . . , r | ∞), H ∗ (F(∞))), k r→0

but lim Hol(Discs(r, . . . , r | ∞), H ∗ (F(∞))) = Hol(Confk (C), H ∗ (F(∞))),

r→0

where Confk (C) is the configuration space of k ordered distinct points in C. Note that if the zi lie in a disc D(0, r), this map provides a map F (r)) mzH∗( : V ⊗k → Hol(Confk (D(0, r)), H ∗ (F(r))). 1 ,...,zk H ∗ (F (∞))

Composing the prefactorization product map mz1 ,...,zk   πn : H ∗ (F(∞)) → V = Vn n

with the map

n

gives a map mz1 ,...,zk : V ⊗k → Hol(Confk (C), V) =



Hol(Confk (C), Vn ).

n

This map does not involve the spaces F(r) anymore, only the space V and its natural completion V. If there is potential confusion, we will refer to this version of the operator product map by mVz1 ,...,zk instead of just mz1 ,...,zk . The vertex operator will, of course, be constructed from the maps mz1 ,...,zk . Consider the map  mz,0 : V ⊗ V → Hol(C× , Vn ), n

where we have restricted the map mVz,w to the locus where w = 0. Since each space Vn is a discrete vector space (i.e., a colimit of finite dimensional vector spaces), we can form the ordinary Laurent expansion of an element in Hol(C× , Vn ) to get a map Lz mVz,0 : V ⊗ V → V[[z, z−1 ]].

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It has the following important property. Lemma 5.3.5 The image of Lz mVz,0 is in the subspace V((z)). Proof The map mVz,0 is S1 -equivariant, where S1 acts on V and C× in the evident way. Therefore, so is Lz mVz,0 . Since every element in V ⊗ V is in a finite sum of the S1 -eigenspaces, the image of Lz mVz,0 is in the subspace of V[[z, z−1 ]] spanned by finite sums of eigenvectors. An element of V[[z, z−1 ]] is in the weight k eigenspace of the S1 action if it is of the form  zk−n vn , n

where vn ∈ Vn . Since Vn = 0 for n $ 0, every such element is in V((z)). The Vertex Algebra Structure Let us now define the structures on V that will correspond to the vertex algebra structure. (i) The vacuum element |0 ∈ V: By assumption, F is a unital prefactorization algebra. Therefore, the commutative algebra F(∅) has a unit element |0 . The prefactorization structure map F(∅) → F(D(0, r)) for any r gives an element |0 ∈ F(r). This element is automatically S1 invariant, and therefore in V0 . (ii) The translation map T : V → V: The structure of a holomorphically ∂ cortranslation prefactorization algebra on F includes a derivation ∂z responding to infinitesimal translation in the (complex) direction z. The ∂ fact that F has a compatible S1 action means that, for all r, the map ∂z maps Fk (r) to Fk−1 (r). Therefore, passing to cohomology, it becomes a ∂ ∂ : Vk → Vk−1 . Let T : V → V be the map that is ∂z on Vk . map ∂z −1 (iii) The state-field map Y : V → End(V)[[z, z ]] : We let Y(v, z)(v ) = Lz mz,0 (v, v ) ∈ V((z)). Note that Y(v, z) is a field in the sense used in the axioms of a vertex algebra, because Y(v, z)(v ) has only finitely many negative powers of z. It remains to check the axioms of a vertex algebra. We need to verify: (i) Vacuum axiom: Y(|0 , z)(v) = v. (ii) Translation axiom: Y(Tv1 , z)(v2 ) = and T |0 = 0.

∂ Y(v1 , z)(v2 ) ∂z

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(iii) Locality axiom: (z1 − z2 )N [Y(v1 , z1 ), Y(v2 , z2 )] = 0 for N sufficiently large. The vacuum axiom follows from the fact that the unit |0 ∈ F(∅), viewed as an element of F(D(0, r)), is a unit for the prefactorization product. The translation axiom follows immediately from the corresponding axiom of prefactorization algebras, which is built into our definition of holomorphic translation invariance: namely, ∂ mz,0 (v1 , v2 ) = mz,0 (∂z v1 , v2 ). ∂z The fact that T |0 = 0 follows from the fact that the derivation ∂z of F gives a derivation of the commutative algebra F(∅), which must therefore send the unit to zero. It remains to prove the locality axiom. Essentially, it follows from the associativity property of prefactorization algebras.

5.3.5 Proof of the Locality Axiom Let us observe some useful properties of the operator product maps mz1 ,...,zk : V ⊗k → Hol(Confk (C), V). First, the map mz1 ,...,zk is S1 -equivariant, where we use the diagonal S1 action on V ⊗k on the left-hand side and on the right-hand side we use the rotation action of S1 on Confk (C) coupled to the S1 action on V coming from F’s S1 -equivariance. The operator product is also Sk -equivariant, where Sk acts on V ⊗k and on Confk (C) in the evident way. It is also invariant under translation, in the sense that for arbitrary vi ∈ V,  mz1 +λ,...,zk +λ (v1 , . . . , vk ) = τλ mz1 ,...,zk (v1 , . . . , vk ) ∈ V where τλ denotes the action of C on V that integrates the infinitesimal translation action of the Lie algebra C. We write  zk mk (v1 , v2 ) Lz mz,0 (v1 , v2 ) = k

mk (v1 , v2 )

∈ V. As we have seen, mk (v1 , v2 ) is zero for k % 0. where The key proposition in proving the locality axiom is the following bit of complex analysis.

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Proposition + Uij ⊂ + Confk (C) be the open subset of configurations + 5.3.6+ Let for which +zj − zi + < +zj − zl + for l = i, j. Then, for (z1 , . . . , zk ) ∈ Uij , we have the following identity in Hol(Uij , V):  mz1 ···zk (v1 , . . . , vk ) = (zi − zj )n mz1 ··· zj ···zk × (v1 , . . . , vi−1 , mn (vi , vj ), . . . ,  vj , . . . , vk ). In particular, the sum on the right-hand side converges. (In this expression, the vj indicate that we skip these entries.) hats  zj and  When k = 3, this identity allows us to expand mz1 ,z2 ,z3 in two different ways. We find that $ (z − z3 )k mz1 ,z3 (mk (v1 , v2 ), v3 ) mz1 ,z2 ,z3 (v1 , v2 , v3 ) =  2 (z1 − z3 )k mz2 ,z3 (v2 , mk (v1 , v3 ))

if |z2 − z3 | < |z1 − z3 | if |z2 − z3 | > |z1 − z3 | .

This formula should be compared to the “associativity” property in the theory of vertex algebras (see, e.g., Theorem 3.2.1 of Frenkel and Ben-Zvi (2004)). Proof By symmetry, we can reduce to the case when i = 1 and j = 2. Since the operator product is induced from the structure maps of H ∗ F, the argument will proceed by analyzing the operator product for discs of small radius and then showing that the relevant property extends to the case where the radii go to zero. Fix a configuration (z1 , . . . , zk ) ∈ Confk (C). There is an ε > 0 such that the closures of every disc D(zi , ε) are disjoint. As we are interested in the region U12 , we can find δ > ε (possibly after shrinking ε) such that D(z1 , ε) ⊂ D(z2 , δ) and the closure of D(z2 , δ) is disjoint from the closure of D(zm , ε) for m > 2. There is an open neighborhood U  of the configuration (z1 , . . . , zk ) in U12 where these conditions hold. For configurations in U  , the axioms of a prefactorization algebra tell us that the following associativity condition holds: mz1 ,z2 ,...,zk (v1 , . . . , vk ) = mz2 ,z3 ,...,zk (mz1 ,z2 (v1 , v2 ), v3 , . . . , vk ).

(†)

Here, we view mz1 ,z2 (v1 , v2 ) as an element of H ∗ (F(D(z2 , δ))) that depends holomorphically on points (z1 , z2 ) that lie in the open set where D(z1 , ε) is disjoint from D(z2 , ε) and contained in D(z2 , δ). By translating z2 to 0, we identify H ∗ (F(D(z2 , δ))) with H ∗ (F(δ)). This associativity property is an immediate consequence of the axioms of a prefactorization algebra.

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By taking ε and δ sufficiently small, we can cover U12 by sets of the form U  . Hence, locally on U12 , we have the associativity we need. We will continue to work with our open U  described earlier. We now assume, without loss of generality, that each vi is homogeneous of weight |vi | under the S1 action on V. Let πk : H ∗ (F(δ)) → Vk be the projection onto the eigenspace Vk . Recall that the algebra D(S1 ) of distributions on S1 acts on F(δ) and so on H ∗ (F(δ)). If this action is denoted by ∗, and if λ ∈ S1 ⊂ C× denotes an element of the circle viewed as a complex number, then πk (f ) = λ−k ∗ f . Note that S1 -equivariance of the operator product map means that we can write the Laurent expansion of mz,0 (v1 , v2 ) as   zk mk (v1 , v2 ) = π|v1 |+|v2 |−k mz,0 (v1 , v2 ). mz,0 (v1 , v2 ) = k

k

That is, zk mk (v1 , v2 ) is the projection of mz,0 (v1 , v2 ) onto the weight |v1 + v2 |− k eigenspace of V. We want to show that  (z1 − z2 )n mz2 ,z3 ,...,zk (mn (v1 , v2 ), . . . , vk ) = mz1 ,...,zk (v1 , . . . , vk ) (‡) n

where the zi lie in our open U  . This identity is equivalent to showing that  mz2 ,z3 ,...,zk (πn mz1 ,z2 (v1 , v2 ), . . . , vk ) = mz1 ,...,zk (v1 , . . . , vk ). n

Indeed, the Laurent expansion of mz1 ,z2 (v1 , v2 ) and the expansion in terms of eigenspaces of the S1 action on V differ only by a reordering of the sum. Fix v1 , . . . , vk ∈ U  . Define a map :

D(S1 ) α

→ Hol(U  , V) → mz2 ,z3 ,...,zk (α ∗ mz1 ,z2 (v1 , v2 ), v3 , . . . , vk ).

Here α ∗ − refers to the action of D(S1 ) on H ∗ (F(δ)). Note that  is a smooth map. Let δ1 denote the delta-function on S1 with support at the identity 1. The associativity identity (†) of prefactorization algebras implies that (δ1 ) = mz1 ,...,zk (v1 , . . . , vk ). The point here is that δ1 ∗ is the identity on H ∗ (F(δ)).

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To prove the identity (‡), it now suffices to prove that    n  λ = (δ1 ). n∈Z

 In the space D(S1 ) with its natural topology the sum n∈Z λn converges to δ1 , as it is simply the Fourier expansion of the delta function. So, to prove the proposition, it suffices to prove that  is continuous, and not just smooth, δ , V) are endowed with their natural topolowhere the spaces D(S1 ) and Hol(U12 δ gies. (The topology on Hol(Uij , V) is given by saying that a sequence converges if its projection to each Vk converges uniformly, with all derivatives, on compact sets of Uijδ .) The spaces D(S1 ) and Hol(Uijδ , V) both lie in the essential image of the functor from locally convex topological vector spaces to differentiable vector spaces. A result of Kriegl and Michor (1997) tells us that smooth linear maps between topological vector spaces are bounded. Therefore, the map  is a bounded linear map of topological vector spaces. In Lemma B.5.5 in Appendix B we show, using results of Kriegl and Michor (1997), that the space of compactly supported distributions on any manifold has the bornological property, meaning that a bounded linear map from it to any topological vector space is the same as a continuous linear map. It follows that  is continuous, thus completing the proof. As a corollary, we find the following. Corollary 5.3.7 For v1 , . . . , vk ∈ V, mz1 ,...,zk (v1 , . . . , vk ) has finite order poles on every diagonal in Confk (C). That is, for some N sufficiently large, the function  ( (zi − zj )N )mz1 ,...,zk (v1 , . . . , vk ) i 0, then $ C∞ (M, Hol(X, E)) if i = 0, i ∞ 0,∗ H (C (M, (X, E))) = 0 if i > 0. Proof The first statement follows from the second statement. Locally on X, the Dolbeault–Grothendieck lemma tells us that the sheaf 0,∗ (X, E) has no higher cohomology, and the sheaves C∞ (M, 0,i (X, E)) are certainly fine. To prove the second statement, consider the exact sequence 0 → Hol(X, E) → 0,0 (X, E) → 0,1 (X, E) · · · → 0,n (X, E) → 0. It is an exact sequence of Fr´echet spaces. Proposition 3 of section I.1.2 in Grothendieck (1955) tells us that the completed projective tensor product of nuclear Fr´echet spaces is an exact functor, that is, takes exact sequences to exact sequences. Another result of Grothendieck (1952) tells us that for any complete locally convex topological vector space F, C∞ (M, F) is naturally π F, where ⊗ π denotes the completed projective tensor isomorphic to C∞ (M)⊗ product. The result then follows. Proof of the theorem We need to produce an isomorphism of differentiable vector spaces, i.e., an isomorphism of sheaves. In other words, we need to find an isomorphism between their sections on each manifold M, 1 ∼ ∞ C∞ (M, H n ( 0,∗ c (Dr1 ,...,rn ))) = C (M, Hol((P \ Dr1 )

× . . . (P1 × Drn ), O(−1)n )), and this isomorphism must be natural in M. We also need to show that the sections C∞ (M, H i ( 0,∗ c (Dr1 ,...,rn ))) of the other cohomology sheaves, with i < n, are zero. The first thing we prove is that, for any complex manifold X and holomorphic vector bundle E on X, and any open subset U ⊂ X, there is an exact sequence ∞ 0,∗ 0 → C∞ (M, 0,∗ c (U, E)) → C (M, (X, E))

→ C∞ (M, 0,∗ (X \ U, E)) → 0

(‡)

of the sections on any smooth manifold M. We will do this by working with these differentiable cochain complexes 0,∗ (X, E) simply as sheaves on M, not on the whole site of smooth manifolds. To start, we will prove that we have an exact sequence of sheaves on M. As

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these complexes are built out of topological vector spaces, they are fine sheaves on M and hence their global sections will provide the exact sequence (‡). Pick an exhausting family {Ki } of compact subsets of U. Define   res 0,∗ 0,∗ 0,∗ (U, E) = Ker (U, E) − → (U \ K , E) . i Ki This vector space coincides with 0,∗ Ki (X, E), since a section that vanishes outside of Ki in X must vanish outside of U. The sequence 0,∗ 0,∗ 0 → 0,∗ Ki (X, E) → (X, E) → (X \ Ki , E)

is therefore exact, but it is not necessarily exact on the right. Consider the colimit of this sequence as i goes to ∞ in the category of sheaves on M. We then obtain an exact sequence 0,∗ 0,∗ 0 → 0,∗ c (U, E) → (X, E) → (X \ U, E)

as sheaves on M. Now, we claim that this sequence is exact on the right, as sheaves. The point is that, locally on M, every smooth function on M × (X \ U) extends to a smooth function on some M × (X \ Ki ) and by applying a bump function that is 1 on a neighborhood of M × (X \ U) and zero on the interior of Ki , we can extend it to a smooth function on M × X. Let’s apply this exact sequence to the case when X = (P1 )n , U = Dr1 ,...,rn and E = O(−1)n . Note that in this case E is trivialized on U, so that we find, on taking cohomology of the Dolbeault complexes, a long exact sequence of sheaves on M: i 0,∗ 1 n n · · · → H i ( 0,∗ c (Dr1 ,...,rn )) → H ( ((P ) , (O(−1)) ))

→ H i ( 0,∗ ((P1 )n \ Dr1 ,...,rn , O(−1)n )) → · · · . Note that the Dolbeault cohomology of (P1 )n with coefficients in O(−1)n vanishes. Hence, the middle term in this exact sequence vanishes by the preceding proposition, and so we get an isomorphism i−1 ∼ ∞ ( 0,∗ ((P1 )n \Dr1 ,...,rn , O(−1)n ))). C∞ (M, H i ( 0,∗ c (Dr1 ,...,rn ))) = C (M, H

For the purposes of the theorem, we now need to compute the right-hand side. The complex appearing on the right-hand side is the colimit, as ε → 0, of C∞ (M, 0,∗ ((P1 )n \ Dr1 −ε,...,rn −ε , O(−1)n )). Here D indicates the closed disc.  We now have a sheaf on M × (P1 )n \ Dr1 −ε,...,rn −ε that sends an open set U × V to C∞ (U, 0,∗ (V, O(−1)n )). It is a cochain complex of fine sheaves.

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181

We are interested in the cohomology of global sections. We can compute this sheaf cohomology using the local-to-global spectral sequence associated to a ˇ Cech cover of (P1 )n \ Dr1 −ε,...,rn −ε . Consider the cover {Ui }1≤i≤n given by Ui = P1 × · · · × (P1 \ Dri −ε ) × · · · × P1 .  We take the corresponding cover of M × (P1 )n \ Dr1 −ε,...,rn −ε given by the opens {M × Ui }1≤i≤n . Note that, for k < n, we have H ∗ ( 0,∗ (Ui1 ,...,ik , O(−1)n )) = 0, where Ui1 ,...,ik denotes the intersection of all the Uij . The previous proposition implies that we also have H ∗ (C∞ (M, 0,∗ (Ui1 ,...,ik , O(−1)n ))) = 0. The local-to-global spectral sequence then tells us that we have a natural isomorphism identifying H ∗ (C∞ (M, 0,∗ ((P1 )n \ Dr1 −ε,...,rn −ε , O(−1)n ))) with

C∞ (M, Hol(U1,...,n , O(−1)n ))[−n].

That is, all cohomology groups on the left-hand side of this equation are zero, except for the top cohomology, which is equal to the vector space on the right. Note that U1,...,n = (P1 \ Dr1 −ε ) × · · · × (P1 \ Drn −ε ). Taking the colimit as ε → 0, combined with our previous calculations, gives the desired result. Note that this colimit must be taken in the category of sheaves on M. We are also using the fact that sequential colimits commute with formation of cohomology.

5.4.6 A Description of Observables on a Disc This analytic discussion allows us to understand both classical and quantum observables of the βγ system. Let us first define some basic classical observables. Definition 5.4.9 On any disc D(x, r) centered at the point x, let cn (x) denote the linear classical observable cn (x) : γ ∈ 0,0 (D(x, r)) →

1 n (∂ γ )(x). n! z

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Likewise, for n > 0, let bn (x) denote the linear functional bn (x) : β dz ∈ 1,0 (D(x, r)) →

1 (∂ n−1 β)(x). (n − 1)! z

These observables descend to elements of H 0 (Obscl (D(x, r))). Let us introduce some notation to deal with products of these. For a multiindex K = (k1 , . . . , kn ) or L = (l1 , . . . , lm ), we let bK (x) = bk1 (x) · · · bkn (x), cL (x) = cl1 (x) · · · clm (x). Of course these products make sense only if ki > 0 for all i. Note that under the natural S1 action on H 0 (Obscl (D(x, r))), the elements bK (x) and cL (x) are  of weight − |K| and − |L|, where |K| = i ki and similarly for L. Lemma 5.4.10 These observables generate the cohomology of the classical observables in the following sense. (i) For i = 0 the cohomology groups H i (Obscl (D(x, r))) vanish as differentiable vector spaces. (ii) The monomials {bK (x)cL (x)}|K|+|L|=n form a basis for the weight n space H 0 (Obscl (D(x, r)))n . Further, they form a basis in the sense of differentiable vector spaces, meaning that any smooth section f : M → Hn0 (Obscl (D(x, r))) can be expressed uniquely as a sum  f = fKL bK (x)cL (x), |K|+|L|=n

where fKL ∈ C∞ (M) and locally on M all but finitely many of the fKL are zero. (iii) More generally, any smooth section f : M → H 0 (Obscl (D(x, r))) can be expressed uniquely as a sum  f = fKL bk (x)cL (x) that is convergent in a natural topology on C∞ (M, H 0 (Obscl (D(x, r)))), with the coefficient functions fKL satisfying the following properties: (a) Locally on M, fk1 ,...,kn ,l1 ,...,lm = 0 for n + m $ 0.

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(b) For fixed n and m, the sum  −kn −l1 −1 1 fk1 ,...,kn ,l1 ,...,lm z−k · · · w1−lm −1 1 · · · zn w1 K,L

+ + is absolutely convergent when |zi | ≥ r and +wj + ≥ r, in the natural topology on C∞ (M). This lemma gives an explicit description of C∞ (M, H 0 (Obscl (D(x, r)))). Proof We will set x = 0. By definition, Obscl (D(0, r)) is a direct sum, as differentiable cochain complexes,   n n 0,∗ , Obscl (U) = c (D(0, r) , E )[n] n

Sn

where E is the holomorphic vector bundle O ⊕ K on C. It follows from Theorem 5.4.7 that, as differentiable vector spaces, H i ( 0,∗ (D(0, r)n , En ) = 0 unless i = n, and that H n ( 0,∗ (D(0, r)n , En ) = Hol((P1 \ D(0, r)), (O(−1) ⊕ O(−1)dz)n ). Theorem 5.4.7 was stated only for the trivial bundle, but the bundle E is trivial on D(x, r). It is not, however, S1 -equivariantly trivial. The notation O(−1)dz indicates how we have changed the S1 -action on O(−1). It follows that we have an isomorphism, of S1 -equivariant differentiable vector spaces  H 0 ((P1 \ D(0, r))n , (O(−1) ⊕ O(−1)dz)n ))Sn . H 0 (Obscl (D(0, r))) ∼ = n

Under this isomorphism, the observable bK (0)cL (0) goes to the function 1 −l1 −1 −lm −1 n z−k1 dz1 · · · z−k · · · wm . n dzn w1 (2π i)n+m 1 Everything in the statement is now immediate. On H 0 (Obscl (D(0, r))) we use the topology that is the colimit of the topologies on holomorphic functions on (P1 \ D(0, r − ε))n as ε → 0. Lemma 5.4.11 We have H ∗ (Obsq (U)) = H ∗ (Obscl (U))[h] ¯ as S1 -equivariant differentiable vector spaces. This isomorphism is compatible with maps induced from inclusions U → V of open subsets of C. It is not compatible with the prefactorization product map.

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Proof This lemma follows, as explained in Section 4.6 in Chapter 4, from the existence of a Green’s function for the ∂ operator, namely G(z1 , z2 ) =

dz1 − dz2 ∈ E (C) ⊗ E (C) z1 − z2

(5.4.6.1)

where E denotes the complex of fields of the βγ system. We will use the notation bK (x)cL (x) for the quantum observables on D(x, r) that arise from the classical observables discussed above, using the isomorphism given by this lemma. Corollary 5.4.12 The properties listed in Lemma 5.4.10 also hold for quantum observables. As a result, all the conditions of Theorem 5.3.3 are satisfied, so that the structure of prefactorization algebra leads to a C[h]-linear vertex ¯ algebra structure on the space  H ∗ (Obsq (D(0, r)))n V= n

where H ∗ (Obsq (D(0, r)))n indicates the weight n eigenspace of the S1 action. Proof The only conditions we have not yet checked are (i) The inclusion maps H ∗ (Obsq (D(0, r)))n → H ∗ (Obsq (D(0, s)))n for r < s are quasi-isomorphisms, and (ii) the differentiable vector spaces Vn = H ∗ (Obsq (D(0, r)))n are countable colimits of finite-dimensional vector spaces in the category of differentiable vector spaces. Both of these conditions follow immediately from the analog of Lemma 5.4.10 that applies to quantum observables.

5.4.7 The Proof of Theorem 5.4.2 We now finish the proof of the main theorem. We have shown that we obtain some vertex algebra, because we have verified the criteria to apply Theorem 5.3.3. What remains is to exhibit an explicit isomorphism with the βγ vertex algebra. Proof Note that Vh¯ =2π i is the polynomial algebra on the generators bn , cm where n ≥ 1 and m ≥ 0. Also bn , cm have weights −n, −m respectively, under the S1 -action. We define an isomorphism Vh¯ =2π i to W by sending bn to a−n and

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185

cm to a∗−m , and extending it to be an isomorphism of commutative algebras. By the reconstruction theorem, it suffices to calculate Y(b1 , z) and Y(c0 , z). By the way we defined the vertex algebra associated to the prefactorization algebra of quantum observables in Theorem 5.3.3, we have Y(b1 , z)(α) = Lz mz,0 (b1 , α) ∈ Vh¯ =2π i ((z)), where Lz denotes Laurent expansion, and mz,0 : Vh¯ =2π i ⊗ Vh¯ =2π i → V h¯ =2π i is the map associated to the prefactorization product coming from the inclusion of the disjoint discs D(z, r) and D(0, r) into D(0, ∞) (where r can be taken to be arbitrarily small). We are using the Green’s function for the ∂ operator to identify classical and quantum observables. Let us recall how the Green’s function leads to an explicit formula for the prefactorization product. Let E denote the sheaf on C of fields of our theory, so that E (U) = 0,∗ (U, O ⊕ K). It is the sheaf of smooth sections of a graded bundle E on C. Let E ! = E [1] denote the sheaf of smooth sections of the vector bundle E∨ ⊗ ∧2 T ∗ , so that a section is a fiberwise linear functional on E valued in 2-forms. Let E denote the sheaf of distributional sections and Ec denote compactly supported sections. The propagator, or Green’s function, is P=

dz1 ⊗ 1 − 1 ⊗ dz2 . 2π i(z1 − z2 )

It is an element of π E (C) = D(C2 , E  E), E (C)⊗ where D denotes the space of distributional sections. We can also view at as a symmetric and smooth linear map β Ec! (C) = Cc∞ (C2 , (E! )2 ) → C. P : Ec! (C) ⊗

(†)

β denotes the completed bornological tensor product on the category of Here ⊗ convenient vector spaces. Recall that we identify Obscl (U) = Sym(Ec! (U)) = Sym(Ec (U)[1]), where the symmetric algebra is defined using the completed tensor product on the category of convenient vector spaces.

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From P we construct an order two differential operator on this symmetric algebra: ∂P : Obscl (U) → Obscl (U). This operator is characterized by the fact that it is a smooth (or, equivalently, continuous) order two differential operator, vanishes on Sym≤1 , and on Sym2 is determined by the map in (†). For example, if x, y ∈ U, then ∂P (bi (x)cj (y)) =

∂ i−1 ∂ j 1 1 . j i−1 (i − 1)!j! ∂ x ∂ y 2π i(x − y)

To compute this, note that bi (x)cj (y) is quadratic and we can pick a represenβ Ec! (C). We then apply P to obtain the tative for this observable in Ec! (C)⊗ number on the right-hand side. Explicitly, bi (x) and cj (y) are derivatives of delta functions and we apply them to the Green’s function. We can identify Obsq (U) as a graded vector space with Obscl (U)[h]. ¯ The quantum differential is d = d1 + h¯ d2 , a sum of two terms, where d1 is the differential on Obscl (U) and d2 is the differential arising from the Lie bracket in the shifted Heisenberg Lie algebra whose Chevalley–Eilenberg chain complex defines Obsq (U). It is easy to check that [h∂ ¯ P , d1 ] = d2 . This identity follows immediately from the fact that (1, 1)-current ∂P on C2 is the delta current on the diagonal. As we explained in Section 4.6 in Chapter 4, we get an isomorphism of cochain complexes W:

Obscl (U)[h] ¯ → α →

Obsq (U) . eh¯ ∂P α

Further, for U1 , U2 disjoint opens in V, the prefactorization product map cl cl h¯ : Obscl (U1 )[h] ¯ × Obs (U2 )[h] ¯ → Obs (V)[h], ¯

arising from that on Obsq under the identification W, is given by the formula     α h¯ β = e−h¯ ∂P eh¯ ∂P α · eh¯ ∂P β . Here · refers to the commutative product on classical observables. Let us apply this formula to α = b1 (z) and β in the algebra generated by bi (0) and cj (0). First, note that since b1 (z) is linear, h¯ ∂P b1 (z) = 0. Note also

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187

that [∂P , b1 (z)] commutes with ∂P . Thus, we find that   b1 (z) h¯ β = e−h¯ ∂P b1 (z)eh¯ ∂P β = b1 (z)β − [h∂ ¯ P , b1 (z)]β. Note that [∂P , b1 (z)] is an order one operator, and so a derivation. So it suffices to calculate what it does on generators. We find that

+ + 1 1 1 ∂j + [∂P , b1 (z)]cj (0) = 2π i j! ∂ j w z − w +w=0 1 −j−1 z = , 2π i and [∂P , b1 (z)]bj (0) = 0. In other words,

∞

[∂P , b1 (z)] =

1  −j−1 ∂ . z 2π i ∂cj (0) j=0

Note as well that for |z| < r, we can expand the cohomology class b1 (z) in H 0 (Obscl (D(0, r)) as a sum b1 (z) =

∞ 

bn+1 (0)zn .

n=0

Indeed, for a classical field γ ∈ 1hol (D(0, r)) satisfying the equations of motion, we have b1 (z)(γ ) = γ (z) =

∞  n=0

=

∞ 

zn

1 (n) γ (0) n!

zn bn1 (0)(γ ).

n=0

Putting all these computations together, we find that, for β in the algebra generated by cj (0), bi (0), we have ∞  ∞   ∂ h ¯ z−m−1 bn+1 (0)zn + b1 (z) h¯ β = 2π i ∂cm (0) n=0

m=0

× β ∈ H 0 (Obscl (D(0, r))[h]. ¯

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Thus, if we set h¯ = 2π i, we see that the operator product on the space Vh¯ =2π i matches the one on W if we sent bn (0) to a−n and cn (0) to a∗−n . A similar calculation of the operator product with c0 (z) completes the proof.

5.5 Kac–Moody Algebras and Factorization Envelopes In this section, we will construct a holomorphically translation invariant prefactorization algebra whose associated vertex algebra is the affine Kac–Moody vertex algebra. This construction is an example of the twisted prefactorization envelope construction, which also produces the prefactorization algebras for free field theories (see Section 3.6 in Chapter 3). As we explained there, it is our version of the chiral envelope construction from Beilinson and Drinfeld (2004). Our work here recovers the Heisenberg vertex algebra, the free fermion vertex algebra, and the affine Kac–Moody vertex algebras. These methods can be applied, however, to any dg Lie algebra with an invariant pairing, so there is a plethora of unexplored prefactorization algebras provided by this construction.

5.5.1 The Context The input data are the following: • A Riemann surface . • A Lie algebra g (for simplicity, we stick to ordinary Lie algebras such as sl2 ). • A g-invariant symmetric pairing κ : g⊗2 → C. From these data, we obtain a cosheaf on , g : U → ( 0,∗ c (U) ⊗ g, ∂), where U denotes an open in . Note that g is a cosheaf of dg vector spaces and merely a precosheaf of dg Lie algebras. When κ is nontrivial (though not necessarily nondegenerate), we obtain a −1-shifted central extension on each open: 0,∗ g κ : U  → ( c (U) ⊗ g, ∂) ⊕ C c, where C denotes the locally constant cosheaf on  and c is a central element of cohomological degree 1. The bracket is defined by

  1 ∂α ∧ β κ(X, Y)c, [α ⊗ X, β ⊗ Y]κ := α ∧ β ⊗ [X, Y] − 2π i U

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189

with α, β ∈ 0,∗ (U) and X, Y ∈ g. (These constants are chosen to match with the use of κ for the affine Kac–Moody algebra in the text that follows.) Remark: Every dg Lie algebra g has a geometric interpretation as a formal moduli space Bg, a theme we develop further in Volume 2. This dg Lie algebra g (U) in fact possesses a natural geometric interpretation: it describes “deformations with compact support in U of the trivial G-bundle on .” Equivalently, it describes the moduli space of holomorphic G-bundles on U which are trivialized outside of a compact set. For U a disc, it is closely related to the affine Grassmannian of G. The affine Grassmannian is defined to be the space of algebraic bundles on a formal disc trivialized away from a point, whereas our formal moduli space describes G-bundles on an actual disc trivialized outside a compact set. The choice of κ has the interpretation of a line bundle on the formal moduli problem Bg (U) for each U. In general, −1-shifted central extensions of a dg Lie algebra g are the same as L∞ -maps g → C, that is, as rank-one representations. Rank-one representations of a group are line bundles on the classifying space of the group. In the same way, rank-one representations of a Lie algebra are line bundles on the formal moduli problem Bg. ♦ As explained in Section 3.6 in Chapte 3, we can form the twisted prefactorization envelope of g . Concretely, this prefactorization algebra assigns to an open subset U ⊂ , the complex F κ (U) = C∗ (g κ (U)) = (Sym( 0,∗ c (U) ⊗ g[1])[c], dCE ), where c now has cohomological degree 0 in the Lie algebra homology complex. It is a prefactorization algebra in modules for the algebra C[c], generated by the central parameter. We should therefore think of it as a family of prefactorization algebras depending on the central parameter c. Remark: Given a dg Lie algebra (g, d), we interpret C∗ g as the “distributions with support on the closed point of the formal space Bg.” Hence, our prefactorization algebras F κ () describes the κ-twisted distributions supported at the point in BunG () given by the trivial bundle on . This description is easier to understand in its global form, particularly when  is a closed Riemann surface. Each point of P ∈ BunG () has an associated dg Lie algebra gP describing the formal neighborhood of P. This dg Lie algebra, in the case of the trivial bundle, is precisely the global sections over  of g . For a nontrivial bundle P, the Lie algebra gP is also global sections of a natural cosheaf, and we can apply the enveloping construction to this cosheaf

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to obtain a prefactorization algebras. By studying families of such bundles, we recognize that our construction C∗ gP should recover differential operators on BunG (). When we include a twist κ, we should recover κ-twisted differential operators. When the twist is integral, the twist corresponds to a line bundle on BunG () and the twisted differential operators are precisely differential operators for that line bundle. It is nontrivial to properly define differential operators on the stack BunG (), and we will not attempt it here. At the formal neighbourhood of a point, however, there are no difficulties and our statements are rigorous. ♦

5.5.2 The Main Result Note that if we take our Riemann surface to be C, the prefactorization alge∂ is bra F κ is holomorphically translation invariant because the derivation ∂z ∂ homotopically trivial, via the homotopy given by ∂dz . It follows that we are in a situation where we might be able to apply Theorem 5.3.3. The main result of this section is the following. Theorem 5.5.1 The holomorphically translation invariant prefactorization algebra F κ on C satisfies the conditions of Theorem 5.3.3, and so defines a vertex algebra. This vertex algebra is isomorphic to the affine Kac–Moody vertex algebra. Before we can prove this statement, we of course need to describe the affine Kac–Moody vertex aigebra. Recall that the Kac–Moody Lie algebra is the central extension of the loop algebra Lg = g[t, t−1 ], 0 → C · c → gκ → Lg → 0. As vector spaces, we have  gκ = g[t, t−1 ] ⊕ C · c, and the Lie bracket is given by the formula

,  [f (t) ⊗ X, g(t) ⊗ Y]κ := f (t)g(t) ⊗ [X, Y] + f ∂g κ(X, Y) for X, Y ∈ g and f , g -∈ C[t, t−1 ]. Here, c has cohomological degree 0 and is central. The notation denotes an algebraic -version of integration around the unit circle the residue pairing), and so tn dt = 2π iδn,−1 . In particular, - (aka we have tn ∂tm = 2π imδm+n,0 . Observe that g[t] is a sub-Lie algebra of the Kac–Moody algebra. The vacuum module W for the Kac–Moody algebra is the induced representation from the trivial rank one representation of g[t]. The induction-restriction adjunction

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191

provides a natural map C → W of g[t]-modules, where C is the trivial representation of g[t]. Let |0 ∈ W denote the image of 1 ∈ C. There is a useful, explicit description of W as a vector space. Consider the sub Lie algebra C · c ⊕ t−1 g[t−1 ] of  gκ , which is complementary to g[t]. The action of this Lie algebra on the vacuum element |0 ∈ W gives a canonical isomorphism U(C · c ⊕ t−1 g[t−1 ]) ∼ =W as vector spaces. The vacuum module W is also a C[c] module in a natural way, because C[c] is inside the universal enveloping algebra of  gκ . Definition 5.5.2 The Kac–Moody vertex algebra is defined as follows. It is a vertex algebra structure over the base ring C[c] on the vector space W. (Working over the base ring C[c] simply means all maps are C[c]-linear.) By the reconstruction theorem 5.3.4, to specify the vertex algebra structure it suffices to specify the state-field map on a subset of elements of W that generate all of W (in the sense of the reconstruction theorem). The following state-field operations define the vertex algebra structure on W. (i) The vacuum element |0 ∈ W is the unit for the vertex algebra, that is, Y(|0 , z) is the identity. gκ , we have an element X |0 ∈ W. We declare that (ii) If X ∈ t−1 g ⊂   Xn z−1−n Y(X |0 , z) = n∈Z

where Xn = phisms of W.

tn X

∈ gκ , and we are viewing elements of  gκ as endomor-

5.5.3 Verification of the Conditions to Define a Vertex Algebra We need to verify that F κ satisfies the conditions listed in Theorem 5.3.3 guaranteeing that we can construct a vertex algebra. Our situation here is entirely parallel to that of the βγ system, so we will be brief. The first thing to check is that the natural S1 action on F κ (D(0, r)) extends to an action of the algebra D(S1 ) of distributions on the circle, which is easy to see by the same methods as with βγ . Next, we need to check that, if Fnκ (D(0, r)) denotes the weight n eigenspace for the S1 -action, then the following properties hold. (i) The inclusion Fnκ (D(0, r)) → Fnκ (D(0, s)) for r < s is a quasiisomorphism. (ii) For n $ 0, the cohomology H ∗ (Flκ (D(0, r)) vanishes as a sheaf on the site of smooth manifolds.

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(iii) The differentiable vector spaces H ∗ (Fnκ (D(0, r)) are countable sequential colimits of finite-dimensional vector spaces in DVS. Note that

  (D(0, r), g)[1] ⊕ C · c F κ (D(0, r)) = Sym 0,∗ c

with differential dCE the Chevalley–Eilenberg differential for gC κ . Give κ F (D(0, r) an increasing filtration, by degree of the symmetric power. This filtration is compatible with the action of S1 and of D(S1 ). In the associated graded, the differential is just that from the differential ∂ on 0,∗ c (D(0, r)). It follows that there is a spectral sequence of differentiable cochain complexes   H ∗ Gr∗ Flκ (D(0, r)) ⇒ H ∗ Flκ (D(0, r)) . The analytic results we proved in Section 5.4.5 concerning  compactly sup- ported Dolbeault cohomology immediately imply that H ∗ Gr∗ Flκ (D(0, r)) satisfy properties (1)−(3) in the preceding. It follows that these properties also hold for H ∗ (Flκ (D(0, r)).

5.5.4 Proof of the Theorem Let us now prove that the vertex algebra associated to Fκ is isomorphic to the Kac–Moody vertex algebra. The proof will be a little different from the proof of the corresponding result for the βγ system. We first prove a statement concerning the behavior of the prefactorization algebra Fκ on annuli. Consider the radial projection map ρ : C× z

→ R>0 , → |z| .

We define the pushforward prefactorization algebra ρ∗ Fκ on R>0 that assigns to any open subset U ⊂ R>0 , the cochain complex Fκ (ρ −1 (U)). In particular, this prefactorization algebra assigns to an interval (a, b), the space Fκ (A(a, b)), where A(a, b) indicates the annulus of those z with a < |z| < b. The product map for ρ∗ F associated to the inclusion of two disjoint intervals in a larger one arises from the product map for F from the inclusion of two disjoint annuli in a larger one. Recall from the example in Section 3.1.1 in Chapter 3 that any associative algebra A gives rise to a prefactorization algebra Afact on R which assigns to the interval (a, b) the vector space A and whose product map is the multiplication in A. The first result we will show is the following.

5.5 Kac–Moody Algebras and Factorization Envelopes

193

Proposition 5.5.3 There is an injective map of C[c]-linear prefactorization algebras on R>0 i : U( gκ )fact → H ∗ (ρ∗ F κ )

whose image is a dense subspace. This map i is controlled by the following property. Observe that, for every open subset U ⊂ C, the subspace of linear elements 0,∗ κ 0,∗ c (U, g)[1] ⊕ C · c ⊂ C∗ ( c (U, g) ⊕ C · c[1]) = F (U)

is, in fact, a subcomplex. Applying this fact to U = ρ −1 (I) for an interval I ⊂ R>0 and taking cohomology, we obtain a linear map −1 0 κ H 1 ( 0,∗ c (ρ (I))) ⊗ g) ⊕ C · c → H (F (I)).

Note that we have a natural identification −1 1 −1 ∨ H 1 ( 0,∗ c (ρ (I))) = hol (ρ (I)) ,

where ∨ indicates continuous linear dual. Thus, the linear elements of H ∗ F κ , which generate the factorization algebra in a certain sense that we’ll identify below, are g-valued linear functionals on holomorphic 1-forms. The following notation will be quite useful. Fix a circle {|z| = r} where r ∈ I. Performing a contour integral around this circle defines a linear function −1 on 1hol (ρ −1 (I)), and so an element of H 1 ( 0,∗ c (ρ (I))). Denote it by φ(1). Likewise, for each integer n, performing a contour integral against zn around −1 n this circle defines an element of H 1 ( 0,∗ c (ρ (I))) that we call φ(z ). The map i : gκ → H ∗ (F κ (ρ −1 (I))) constructed by the theorem factors through the map i:

 gκ Xzn c

→ → →

−1 H 1 ( 0,∗ c (ρ (I))) ⊗ g ⊕ C · c . φ(zn ) ⊗ X c

Note that Cauchy’s integral theorem ensures that, in a sense, it does not matter what circle we use for the contour integrals that define φ. Hence, this map should produce a map from the universal enveloping algebra of  gκ , viewed as ∗ κ a locally constant prefactorization algebra, to H F . Proof In Chapter 3, Section 3.4, we showed how the universal enveloping algebra of any Lie algebra a arises as a prefactorization envelope. Let U fact (a)

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denote the prefactorization algebra on R that assigns to an interval I, the cochain complex U(a)fact (I) = C∗ ( ∗c (I, a)). We showed that the cohomology of U(a)fact is locally constant and corresponds to the ordinary universal enveloping algebra Ua. Let us apply this construction to a =  gκ . To prove the theorem, we will produce a map of prefactorization algebras on R>0 i : U( gκ )fact → ρ∗ F κ . Since both sides are defined as the Chevalley–Eilenberg chains of local Lie algebras, it suffices to produce such a map at the level of dg Lie algebras. We will produce such a map in a homotopical sense. To be explicit, we will introduce several precosheaves of Lie algebras and quasi-isomorphisms "

"

gκ −→ L1  L1 −→ L2 . ∗c ⊗ 

(†)

In the middle, between L1 and L1 , there will be a cochain homotopy equivalence. At the level of cohomology, we then obtain an isomorphism ∼ =

 gκ ) −→ H ∗ L2 gκ = H ∗ ( ∗c ⊗  that proves the theorem. Let us remark on a small but important point. These precosheaves will satisfy a stronger condition that ensures the functor C∗ of Chevalley–Eilenberg chains produces a prefactorization algebra. Consider the symmetric monoidal category whose underlying category is dg Lie algebras but whose symmetric monoidal structure is given by direct sum as cochain complexes. (The coproduct in dg Lie algebras is not this direct sum.) All four precosheaves from the sequence (†) are prefactorization algebras valued in this symmetric monoidal category. In general, a precosheaf L of dg Lie algebras is such a prefactorization algebra if it has the property that for any two disjoint opens U, V in W, the elements in L(W) coming from L(U) commute with those coming from L(V). We now describe these prefactorization dg Lie algebras. Let L1 be the prefactorization dg Lie algebras on R that assigns to an interval I the dg Lie algebra   L1 (I) = ∗c (I) ⊗ g[z, z−1 ] ⊕ C · c[−1]. Thus, L1 is a central extension of ∗c ⊗ g[z, z−1 ]. The cocycle defining the central extension on the interval I is 

 

, αXzn ⊗ βYzm → α ∧ β κ(X, Y) z n ∂z z m , I

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195

where α, β ∈ ∗c (U). Note that there is a natural map of such prefactorization dg Lie algebras j : ∗c ⊗  gκ → L 1 that is the identity on ∗c ⊗ g[z, z−1 ] but on the central element is given by

  j(αc) = α c, I

∗c (I).

where α ∈ This map is clearly a quasi-isomorphism when I is an interval. Hence, we have produced the first map in the sequence (†). It follows immediately that the map gκ )fact = C∗ ( ∗c ⊗  gκ ) → C∗ L1 C∗ j : U( is a map of prefactorization algebras that is a quasi-isomorphism on intervals. Therefore, the cohomology prefactorization algebra of C∗ L1 assigns to an interval U( gκ ), and the prefactorization product is just the associative product on this algebra. Let L2 be the prefactorization dg Lie algebra ρ∗ gC κ . In other words, it assigns to an interval I the dg Lie algebra   −1 (ρ (I)) ⊗ g ⊕ C · c[−1], L2 (I) = 0,∗ c with the central extension inherited from gC κ . Hence, C∗ L2 = ρ∗ F κ , so that the last element in the sequence (†) is what we need for the theorem. Finally, we define L1 to be a central extension of ∗c ⊗ g[z, z−1 ], but where the cocycle defining the central extension is 

 

 

, ∂ αXzn ⊗βYzm → α ∧ β κ(X, Y) αr β , zn ∂z zm +π κ(X, Y)δn+m,0 ∂r I I ∂ where the vector field r ∂r acts by the Lie derivative on the form β ∈ ∗c (I). It is easy to verify that this cochain is closed and so defines a central extension. Note that L1 looks like L1 except with a small addition to the cocycle giving the central extension.

Remark: This central extension, as well as the ones defining L1 and L2 , are local central extensions of local dg Lie algebras in the sense of Definition 3.6.3 of Chapter 3. This concept is studied in more detail in Volume 2. ♦ To prove the proposition, we will do the following.

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(i) Prove that L1 and L1 are homotopy equivalent prefactorization dg Lie algebras. (ii) Construct a map of precosheaves of dg Lie algebras L1 → L2 . For the first point, note that the extra term

 αr

π κ(X, Y)δn+m,0 U

in the coycle for

L1

∂ β ∂r

is an exact cocycle. It is cobounded by the cochain  αrι ∂ β, π κ(X, Y)δn+m,0 U

∂r

where ι ∂ indicates contraction. The fact that this expression cobounds follows ∂r from the Cartan homotopy formula for the Lie derivative of vector fields on differential forms. Since the cobounding cochain is also local, L1 and L1 are homotopy equivalent as prefactorization dg Lie algebras. Now we will produce the desired map  : L1 → L2 . We use the following notation: • r denotes both the coordinate on R>0 and also the radial coordinate in C× , • θ is the angular coordinate on C× , and • X will denote an element of g. We will view  gκ as g[z, z−1 ] ⊕ C · c. On an open I ⊂ R>0 , the map  is f (r)Xzn f (r)dr Xzn c

→ → →

f (r)zn X, 1 iθ n 2 e f (r)z dz X,

c,

with f ∈ 0c (U). It is a map of precosheaves but we need to verify it is a map of dg Lie algebras. Compatibility with the differential follows from the formula ∂ in polar coordinates: for ∂z

 1 iθ ∂ 1 ∂ ∂ = e − . ∂z 2 ∂r ir ∂θ Only the central extension might cause incompatibility with the Lie bracket. Consider the following identity: 

  1 f (r)zn ∂ zm g(r) eiθ dz = 2π imδn+m,0 f (r)g(r) dr 2 ρ −1 (I) I  ∂ + π δn+m,0 f (r)r g(r) dr, ∂r I

5.5 Kac–Moody Algebras and Factorization Envelopes

197

where I ⊂ R>0 is an interval. The expression on the right-hand side gives the central extension term in the Lie bracket on L1 , whereas that on the left is the central extension term for L2 applied to the elements (f (r)Xzn ) and (g(r)dr Yzm ). Applying Chevalley chains, we get a map of prefactorization algebras C∗  : C∗ L1 → C∗ L2 = ρ∗ Fκ . Since C∗ L1 is homotopy equivalent to C∗ L1 (although we have not given an equivalence explicitly), we get the desired map of prefactorization algebras on R>0 : U( gκ )fact → H ∗ (ρ∗ Fκ ). The analytical results about compactly supported Dolbeault cohomology in Section 5.4.5 imply immediately this map has dense image. Finally, we check that this map has the properties discussed just after the statement of the theorem. Let I ⊂ R>0 be an interval. Then, under the isomorphism ∼ =

i : U( gκ ) − → H ∗ (C∗ (L1 (I))), the element Xzn is represented by the element f (r)zn Xdr ∈ L1 (U)[1],

where f has compact support and is chosen so that f (r) dr = 1. To check the desired properties, we need to verify that if α is a holomorphic 1-form on ρ −1 (I), then  , n 1 iθ αf (r)z 2 e dz = αzn . ρ −1 (I)

|z|=1

But this identity follows immediately from Stokes’ theorem and the equation

where h(r) =

r

f (r) 12 eiθ dz = ∂(h(r)),

∞ f (r) dr.

This theorem shows how to relate observables on an annulus to the universal enveloping algebra of the affine Kac–Moody Lie algebra. Recall that the Kac– Moody vertex algebra is the vacuum representation equipped with a vertex algebra structure. The next result will show that the vertex algebra associated to the prefactorization algebra F κ is also a vertex algebra structure on the vacuum representation. More precisely, we will show the following.

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Proposition 5.5.4 Let V=



H ∗ (Fnκ (D(0, ε))

n

be the cohomology of the direct sum of the weight n eigenspaces of the S1 action on Fκ (D(0, ε)). Let A(r, r ) be the annulus {r < |z| < r }. The map i : U( gκ ) → H ∗ (F κ (A(r, r )) constructed in the previous proposition induces an action of U( gκ ) on V. Moreover, there is a unique isomorphism of U( gκ )-modules from V to the vacuum module W, which sends the unit observable 1 ∈ V to the vacuum element |0 ∈ W. Proof If ε < r < r , then the prefactorization product gives a map F κ (D(0, ε)) × F κ (A(r, r )) → F κ (D(0, r )) of cochain complexes. Passing to cohomology, and using the relationship between U( gκ ) and F κ (A(r, r )), we get a map V ⊗ U( gκ ) → H ∗ (F κ (D(0, r ))). Note that on the left-hand side, every element is a finite sum of elements in S1 -eigenspaces. Since the map is S1 -equivariant, its image is the subspace in H ∗ (F(D(0, r ))) that consists of finite sums of S1 -eigenvectors. This subspace is V. We therefore find a map V ⊗ U( gκ ) → V.

(†)

To see that it makes V a module over U( gκ ), combine the following observations. First, the map i : U( gκ ) → H ∗ (F κ (A(r, r )) takes the asosciative product on U( gκ ) to the prefactorization product map H ∗ (F κ (A(r, r )) ⊗ H ∗ (F κ (A(s, s )) → H ∗ (F κ (A(r, s )) for r < r < s < s . Second, the diagram of prefactorization product maps F κ (D(0, ε)) ⊗ F κ (A(r, r )) ⊗ F κ (A(s, s ))

/ F κ (D(0, ε)) ⊗ F κ (A(r, s ))

 F κ (D(0, r )) ⊗ F κ (A(s, s ))

 / F κ (D(0, s ))

commutes. We need to show that this action identifies V with the vacuum representation W. The putative map from W to V sends the vacuum element |0 to the unit

5.5 Kac–Moody Algebras and Factorization Envelopes

199

observable 1 ∈ V. To show that this map is well defined, we need to show that gκ for n ≥ 0. 1 ∈ V is annihilated by the elements zn X ∈  The unit axiom for prefactorization algebras implies that the following diagram commutes: / H ∗ (F κ (A(s, s )) U( gκ )  / H ∗ (F κ (D(0, s )).

 V

The left vertical arrow is given by the action of U( gκ ) on the unit element 1 ∈ V, and the right vertical arrow is the map arising from the inclusion A(r, r ) ⊂ D(0, r ). The bottom right arrow is the inclusion into the direct sum of S1 -eigenspaces, which is injective. The proof of Theorem 5.5.3 gives an explicit representative for the element gκ in F κ (A(s, s )). Namely, let f (r) be a function supported in the interXzn ∈  val (s, s ) and with f (r) dr = 1. Then Xzn is represented by n 1 iθ 2 e f (r)dzz X

 ∈ 0,1 c (A(s, s )) ⊗ g.

We will show that this element is exact when it is viewed as an element of  n 0,1 c (D(0, s )) ⊗ g, so that Xz vanishes when going right and then down in the square above. Set  h(r) =

r

∞

f (t) dt,

so h(r) = 0 for r $ s and h(r) = 1 for r < s. Also, ∂ tells us that coordinate representation of ∂z

∂ ∂r h

= f . The polar-

∂h(r)zn X = 12 eiθ f (r) dzzn X. gκ , where n ≥ 0, act by zero on Thus, we have shown that the elements Xzn ∈  the element 1 ∈ V. We therefore have a unique map of U( gκ -modules from W to V sending |0 to 1. It remains to show that this map is an isomorphism. Note that every object we are discussing is filtered. The universal enveloping algebra U( gκ ) is filtered, gκ ) to be the subspace spanned by products of ≤ i as usual, by setting F i U( elements of  gκ . (This filtration is inherited from the tensor algebra.) Similarly, the space F κ (U) = C∗ ( 0,∗ c (U, g) ⊕ C · c[−1]) is filtered by saying that F i F κ (U) is the subcomplex C≤i . All the maps we have been discussing are compatible with these increasing filtrations.

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The associated graded of U( gκ ) is the symmetric algebra of gκ , and the associated graded of F κ (U) is the appropriate symmetric algebra on 0,∗ c (U, g)[1]⊕ C · c in differentiable cochain complexes. On taking associated graded, the maps Gr U( gκ ) → Gr H ∗ F κ (A(s, s )), Gr H ∗ (F κ (A(s, s )) → Gr H ∗ F κ (D(0, s )) are maps of commutative algebras. It follows that the map   Gr U( gκ ) → Gr V ⊂ H ∗ Sym 0,∗ c (D(0, s), g)[1] ⊕ C · c is a map of commutative algebras. Moreover, Gr V is the direct sum of the S1 eigenspaces in the space on the right side of this map. The direct sum of the S1 eigenspaces in H 1 ( 0,1 (D(0, s)) is naturally identified with z−1 C[z−1 ]. Thus, we find the associated graded of the map U( gκ ) to V is a map of commutative algebras     Sym g[z, z−1 ] [c] → Sym z−1 g[z−1 ] [c].

We have already calculated that on the generators of the commutative algebra, it arises from the natural projection map g[z, z−1 ] ⊕ C · c → z−1 g[z−1 ] ⊕ C · c. It follows immediately that the map Gr W → Gr V is an isomorphism, as desired. To complete the proof that the vertex algebra associated to the prefactorization algebra F κ is isomorphic to the Kac–Moody vertex algebra, we need to identify the operator product expansion map Y : V ⊗ V → V((z)). Recall that this map is defined, in terms of the prefactorization algebra F κ , as follows. Consider the map mz,0 : V ⊗ V → H ∗ (F κ (D(0, ∞)) defined by restricting the prefactorization product map H ∗ (F κ (D(z, ε)) × H ∗ (F κ (D(0, ε)) → H ∗ (F κ (D(0, ∞)) to the subspace V ⊂ H ∗ (F κ (D(z, ε)) = H ∗ (F κ (D(0, ε)).

5.5 Kac–Moody Algebras and Factorization Envelopes

201

Composing the map mz,0 with the map H ∗ (F κ (D(0, ∞)) → V =



Vk

k

given by the product of the projection maps to the S1 -eigenspaces, we get a map mz,0 : V ⊗ V → V. This map depends holomorphically on z. The operator product map is obtained as the Laurent expansion of mz,0 . Our aim is to calculate the operator product map and identify it with the vertex operator map in the Kac–Moody vertex algebra. We use the following gκ . We denote the action of  gκ on V by the notation. If X ∈ g, let Xi = zi X ∈  symbol ·. Then we have the following. Proposition 5.5.5 For all v ∈ V, mz,0 (X−1 · 1, v) =



z−i−1 (Xi · v) ∈ V

i∈Z

where the sum on the right-hand side converges. Before we prove this proposition, let us observe that it proves our main result. Corollary 5.5.6 This isomorphism of U( gκ )-modules from the vacuum representation W to V is an isomorphism of vertex algebras, where V is given the vertex algebra structure arising from the prefactorization algebra F κ , and W is given the Kac–Moody vertex algebra structure defined in 5.5.2. Proof This corollary follows immediately from the reconstruction theorem 5.3.4. Proof of the proposition The inclusion of a disc into an annulus induces a structure map of the prefactorization algebra. Each element v of V thus determines an element of the prefactorization algebra on that annulus, and hence a map from V to V. The formula will follow by applying our earlier analysis of the structure maps. gκ denote Xzi for X ∈ g. As we explained in the As before, we let Xi ∈  discussion following Theorem 5.5.3, we can view the element X−1 · 1 ∈ V

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as being represented at the cochain level by X times the linear functional 1hol (D(0, s)) α

→ C → |z|=r z−1 α.

Cauchy’s theorem tells us that this linear functional sends h(z) dz, when h is holomorphic, to 2π ih(0). Now fix z0 ∈ A(s, s ), and let ιz0 : V → H ∗ (F κ (A(s, s ))) denote the map arising from the restriction to V of the structure map H ∗ (F κ (D(0, ε))) = H ∗ (F κ (D(z0 , ε))) → H ∗ (F κ (A(s, s ))) arising from the inclusion of the disc D(z0 , ε) into the annulus A(s, s ). It is clear from the definition of mz0 ,0 and the axioms of a prefactorization algebra that the following diagram commutes: V ⊗ V ⊗ Id

mz0 ,0

/V O

ιz0

 H ∗ (F κ (A(s, s )) ⊗ V

/ H ∗ (F κ (D(0, ∞))).

Here the bottom right arrow is the restriction to V of the prefactorization structure map H ∗ (F κ (A(s, s )) ⊗ H ∗ (F κ (D(0, ε)) → H ∗ (F κ (D(0, ∞)). It therefore suffices to show that  ιz0 (X−1 · 1) = Xi z0−i−i ∈ H ∗ (F κ (A(s, s ))) i∈Z

where we view Xi ∈  gκ as elements of H 0 (F κ (A(s, s ))) via the map U( gκ ) → H 0 (F κ (A(s, s ))) constructed in Theorem 5.5.3. It is clear from the construction of F κ that ιz0 (X−1 · 1) is in the image of the natural map  κ  0,∗ c (A(s, s )) ⊗ g → F (A(s, s )).  Recall that the cohomology of 0,∗ c (A(s, s )) is the linear dual of the space of holomorphic 1-forms on the annulus A(s, s ). The element ιz0 (X−1 · 1) can thus

5.5 Kac–Moody Algebras and Factorization Envelopes

203

be represented by the continuous linear map 1hol (D(0, s))

→

α

→

g

 |z|=r

 z−1 α X.

Similarly, the elements Xi ∈ H ∗ (F κ (A(s, s ))) are represented by the linear maps 

 i z h(z)dz X. h(z)dz → |z|=r

It remains to show that, for all holomorphic functions h(z) on the annulus A(s, s ), we have   , i −i−1 z0 z h(z) dz . 2π ih(z0 ) = i∈Z

|z|=s+ε

This equality is proved using Cauchy’s theorem. We know , , h(z) h(z) dz − dz. 2π ih(z0 ) = |z|=s+ε z − z0 |z|=s −ε z − z0 Expanding (z − z0 )−1 in the regions when |z| < |z0 | (relevant for the first integral) and when |z| > |z0 | (relevant for the second integral) gives the desired expression.

PA RT III Factorization algebras

6 Factorization Algebras: Definitions and Constructions

Our definition of a prefactorization algebra is closely related to that of a precosheaf or of a presheaf. Mathematicians have found it useful to refine the axioms of a presheaf to those of a sheaf: a sheaf is a presheaf whose value on a large open set is determined, in a precise way, by values on arbitrarily small subsets. In this chapter we describe a similar “descent” axiom for prefactorization algebras. We call a prefactorization algebra satisfying this axiom a factorization algebra. After defining this axiom, our next task is to verify that the examples we have constructed so far, such as the observables of a free field theory, satisfy it. This we do in Sections 6.5 and 6.6. Philosophically, our descent axiom for factorization algebras is important: a prefactorization algebra satisfying descent (i.e., a factorization algebra) is built from local data, in a way that a general prefactorization algebra need not be. However, for many practical purposes, such as the applications to field theory, this axiom is often not essential. A reader with little taste for formal mathematics could thus skip this chapter and the next and still be able to follow the rest of this book.

6.1 Factorization Algebras A factorization algebra is a prefactorization algebra that satisfies a local-toglobal axiom. This axiom is the analog of the gluing axiom for sheaves; it expresses how the values on big open sets are determined by the values on small open sets. We thus begin by reviewing the notion of a (co)sheaf, with more background and references available in Appendix A.5. Next, we introduce Weiss covers, which are the type of covers appropriate for factorization algebras. Finally, we give the definition of factorization algebras, 207

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Factorization Algebras: Definitions and Constructions

6.1.1 Sheaves and Cosheaves In order to motivate our definition of factorization algebra, let us recall the sheaf axiom and then work out its dual, the cosheaf axiom. Let M be a topological space and let Opens(M) denote the poset category of open subsets of a space M. A presheaf A on a topological space M with values in a category C is a functor from Opens(M)op to C. Given an open cover {Ui | i ∈ I} of an open set U ⊂ M, there is a canonical map  A(U) → A(Ui ) i

given by taking the product of the structure maps A(U) → A(Ui ). A presheaf A is a sheaf if for every U and for every open cover of U, this map equalizes the pair of maps   A(Ui ) ⇒ A(Ui ∩ Uj ) i

i,j

given by the two inclusions Ui ∩ Uj → Ui and Ui ∩ Uj → Uj . That is, A(U) is the limit of that diagram. In words, this condition means that a section s of A on U can be described as sections si on each element of the cover that agree on the overlaps, i.e., si |Ui ∩Uj = sj |Ui ∩Uj . We now dualize this definition. A precosheaf  on M with values in a category C is a functor from Opens(M) to C. It is a cosheaf if for every open cover of U, the canonical map given by taking the coproduct of the structure map (Uk ) → (U), . (Uk ) → (U), k

coequalizes the pair of maps . . (Ui ∩ Uj ) ⇒ (Uk ) i,j

k

given by the two inclusions Ui ∩ Uj → Ui and Ui ∩ Uj → Uj . That is, (U) is the colimit of that diagram. Thus, a cosheaf with values in C can be understood as a sheaf with values in C op . In this book, our target categories are linear in nature, so that C is the category of vector spaces or of dg vector spaces or of cochain complexes with values in some additive category, such as differentiable vector spaces. In such a setting, we can write the cosheaf axiom in an equivalent but slightly different way. A precosheaf  on M with values in, say, vector spaces is a cosheaf if,

6.1 Factorization Algebras

209

for every open cover of any open set U, the sequence   (Ui ∩ Uj ) → (Uk ) → (U) → 0 i,j

k

is exact, where the first map is the difference of the structure maps for the two inclusions. In the remainder of this book, we phrase our gluing axioms in this style, as we always restrict to linear target categories.

6.1.2 Weiss Covers For factorization algebras, we require our covers to be fine enough that they capture all the “multiplicative structure” – the structure maps – of the underlying prefactorization algebra. In fact, a factorization algebra will be a cosheaf with respect to this modified notion of cover. Definition 6.1.1 Let U be an open set. A collection of open sets U = {Ui | i ∈ I} is a Weiss cover of U if for any finite collection of points {x1 , . . . , xk } in U, there is an open set Ui ∈ U such that {x1 , . . . , xk } ⊂ Ui . The Weiss covers define a Grothendieck topology on Opens(M), the poset category of open subsets of a space M. We call it the Weiss topology of M. (This notion was introduced in Weiss (1999) and it is explained nicely and further developed in Boavida de Brito and Weiss (2013).) Remark: A Weiss cover is certainly a cover in the usual sense, but a Weiss cover typically contains an enormous number of opens. It is a kind of “exponentiation” of the usual notion of cover, because a Weiss cover is well suited to studying all configuration spaces of finitely many points in U. For instance, given a Weiss cover U of U, the collection {Uin | i ∈ I} provides a cover in the usual sense of U n ⊂ M n for every positive integer n. ♦ Example: For a smooth n-manifold M, there are several simple ways to construct a Weiss cover for M. One construction is simply to take the collection of open sets in M diffeomorphic to a disjoint union of finitely many copies of the open n-disc. Alternatively, the collection of opens {M \ q | q ∈ M} is a Weiss cover of M. One can also produce a Weiss cover of a manifold with countably many elements. Fix a Riemannian metric on M. Pick a collection of points {qn ∈ M | n ∈ N} such that the union of the unit “discs” D1 (qn ) = {q ∈ M | d(qn , q) < 1} covers M. (Such an open may not be homeomorphic to a disc if the injectivity radius of qn is less than 1.) Consider the

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Factorization Algebras: Definitions and Constructions

collection of all “discs” D = {D1/m (qn ) | m ∈ N, n ∈ N}. The collection of all finite, pairwise disjoint union of elements of D is a countable Weiss cover. ♦ The preceding examples suggest the following definition. Definition 6.1.2 A cover U = {Uα } of M generates the Weiss cover V if every open V ∈ V is given by a finite disjoint union of opens Uα from U.

6.1.3 Strict Factorization Algebras The value of a factorization algebra on U is determined by its behavior on a Weiss cover, just as the value of a cosheaf on an open set U is determined by its value on any cover of U. Throughout, the target multicategory C will be linear in nature (e.g., vector spaces or cochain complexes of differentiable vector spaces). Definition 6.1.3 A prefactorization algebra on M with values in a multicategory C is a factorization algebra if it has the following property: for every open subset U ⊂ M and every Weiss cover {Ui | i ∈ I} of U, the sequence   F(Ui ∩ Uj ) → F(Uk ) → F(U) → 0 i,j

k

is exact. That is, F is a factorization algebra if it is a cosheaf with respect to the Weiss topology. A factorization algebra is multiplicative if, in addition, for every pair of disjoint open sets U, V ∈ M, the structure map mU,V UV : F(U) ⊗ F(V) → F(U  V) is an isomorphism. To summarize, a multiplicative factorization algebra satisfies two conditions: an axiom of (co)descent and a multiplicativity axiom. The descent axiom says that it is a cosheaf with respect to the Weiss topology, and the multiplicativity axiom says that its value on finite collections of disjoint opens factors into a tensor product of the values on each open. Remark: It is typically not easy to verify directly that a prefactorization algebra is a factorization algebra. In the locally constant case discussed in Section 6.4, we discuss a theorem that exhibits many more examples, including the prefactorization algebra on the real line arising from an associative algebra. Later in this chapter, we exhibit a large class of examples explicitly, arising from

6.1 Factorization Algebras

211

the enveloping construction for local Lie algebras. The next chapter provides analogs of methods familiar from sheaf theory, such as pushforwards and pullbacks and extension from a basis. ♦

ˇ 6.1.4 The Cech Complex and Homotopy Factorization Algebras If our target multicategory has a notion of weak equivalence – such as quasi-isomorphism between cochain complexes – then there is a natural modification of the notion of cosheaf. It arises from the fact that one should work with a corrected notion of colimit that takes into account that some objects are weakly equivalent, even if there is no strict equivalence between them. In other words, we need to work with the notion of colimit appropriate to ∞-categories. We will use the term homotopy colimit for this notion. (Appendix C.5 contains an extensive discussion in the context relevant to us: a category of cochain complexes in an additive category.) One then revises the cosheaf gluing axiom by using homotopy colimits, leading to the notion of a homotopy cosheaf. After describing these, we define homotopy factorization algebras. Remark: Our target category will always be unbounded cochain complexes with values in some additive category C, and we want to treat quasi-isomorphic cochain complexes as equivalent. In other words, we want to treat Ch(C) as an ∞-category, and there are several approaches that are, in a precise sense, equivalent. When C is a Grothendieck Abelian category – as is the case for differentiable vector spaces DVS – this situation is well developed, and much of Appendix C.5 is devoted to understanding it. We discuss there a stable model category, a pretriangulated dg category, a simplicially enriched category, and a stable quasicategory associated to Ch(C), and how to transport constructions and statements between all these approaches. (We rely heavily on and summarize the approach in Lurie (2016).) In light of the results explained there, here we will write down explicit cochain complexes that describe the homotopy colimits of the diagrams we need. Readers familiar with higher categories will recognize how to generalize these definitions but we do not pursue such generality here. Work in that direction can be found in Ayala and Francis (2015) or chapter 5 of Lurie (2016). ♦ Let us provide the definition of a homotopy cosheaf. (See Appendix A.5.3 for more discussion.) Let  be a precosheaf on M with values in cochain complexes in an additive category C. Let U = {Ui | i ∈ I} be a cover of some open subset U of M. Consider the cochain complex with values in cochain

212

Factorization Algebras: Definitions and Constructions

complexes whose −kth term is 

(Uj0 ∩ · · · ∩ Ujk ),

j0 ,...,jk ∈I

ji all distinct

and note that this term has an “internal” differential arising from . The “external” differential is the alternating sum of the structure maps arising from incluˇ sion of k + 1-fold intersections into k-fold intersections. The Cech complex of U with coefficients in  is the totalization of this double complex: ⎛ ⎞ ˇ C(U, ) =

∞  ⎜ ⎜ ⎝ k=0

 j0 ,...,jk ∈I

⎟ (Uj0 ∩ · · · ∩ Ujk )[k]⎟ ⎠.

ji all distinct

There is a natural map from the 0th term (of the double complex) to (U) ˇ given by the sum of the structure maps (Ui ) → (U). Thus, the Cech complex possesses a natural map to (U). We say that  is a homotopy cosheaf ˇ if the natural map from the Cech complex to (U) is a quasi-isomorphism for every open U ⊂ M and every open cover of U. ˇ ˇ Remark: This Cech complex is simply the dual of the Cech complex for sheaves ˇ of cochain complexes. Note that this Cech complex is also the normalized cochain complex arising from a simplicial cochain complex, where we evaluate  on the simplicial space U• associated to the cover U. In more sophistiˇ cated language, the Cech complex describes the homotopy colimit of  on this simplicial diagram. ♦. We can now define our main object of interest. Let C be a multicategory whose underlying category is a Grothendieck abelian category. Then there is a natural multicategory whose underlying category is Ch(C), the category of cochain complexes in C in which the weak equivalences are quasiisomorphisms. Definition 6.1.4 A homotopy factorization algebra is a prefactorization algebra F on X valued in Ch(C), with the property that for every open set U ⊂ X and Weiss cover U of U, the natural map ˇ C(U, F) → F(U) is a quasi-isomorphism. That is, F is a homotopy cosheaf with respect to the Weiss topology. If C is a symmetric monoidal category, then we can formulate a multiplicativity condition, as follows.

6.1 Factorization Algebras

213

Definition 6.1.5 A homotopy factorization algebra F is multiplicative if for every pair U, V of disjoint open subsets of X, the structure map F(U) ⊗ F(V) → F(U  V) is a quasi-isomorphism. Remark: In light of the weak equivalences, the strict notion of multiplicative factorization algebra is not appropriate for the world of cochain complexes. Whenever we refer to a factorization algebra in cochain complexes, we mean a homotopy factorization algebra. ♦ Because we will work primarily with differentiable cochain complexes, we introduce the following terminology. Definition 6.1.6 A differentiable factorization algebra is a factorization algebra valued in the multicategory of differentiable cochain complexes. Remark: Although the multiplicativity axiom does not directly extend to the setting of multicategories, we can examine the bilinear structure map mU,V UV ∈ Ch(DVS)(F(U), F(V) | F(U  V)) for two disjoint opens U and V, and we can ask about the image of mU,V UV . For almost all the examples constructed in this book, the differentiable vector spaces that appear are convenient vector spaces, which have a natural topology. In our constructions, one can typically see that the image of the structure map is dense with respect to this natural topology on F(U  V), which is a cousin of the multiplicativity axiom. In practice, though, it is the descent axiom that is especially important for our purposes. ♦

6.1.5 Weak Equivalences The notion of weak equivalence, or quasi-isomorphism, on cochain complexes induces such a notion on homotopy factorization algebras. Definition 6.1.7 Let F, G be homotopy factorization algebras valued in Ch(C), as above. A map φ : F → G of homotopy factorization algebras is a weak equivalence if, for all open subsets U ⊂ M, the map F(U) → G(U) is a quasi-isomorphism of cochain complexes. We now provide an explicit criterion for checking weak equivalences, using the notion of a factorizing basis (see Definition 7.2.1). Lemma 6.1.8 A map F → G between differentiable factorization algebras is a weak equivalence if and only if, for every factorizing basis U of X and every

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Factorization Algebras: Definitions and Constructions

U in U, the map F(U) → G(U) is a weak equivalence. Proof For any open subset V ⊂ X, let UV denote the Weiss cover of V generated by all open subsets in U that lie in V. By the descent axiom, the map ˇ V , F) → F(V) C(U is a weak equivalence, and similarly for G. Thus, it suffices to check that the map ˇ V , G) ˇ V , F) → C(U C(U ˇ is a weak equivalence, for the following reason. Every Cech complex has a natural filtration by number of intersections. We thus obtain of spectral sequences ˇ from the map of Cech complexes. The fact that the maps F(U1 ∩ · · · ∩ Uk ) → G(U1 ∩ · · · ∩ Uk ) are all weak equivalences implies that we have a quasi-isomorphism on the first page of the map of spectral sequences, so our original map is a quasiisomorphism.

6.1.6 The (Multi)category Structure on Factorization Algebras Every factorization algebra is itself a prefactorization algebra. Thus we define the category of factorization algebras FA(X, C) to be the full subcategory whose objects are the factorization algebras. We have also described in Section 3.1.5 in Chapter 3 the symmetric monoidal category of prefactorization algebras. In brief, the tensor product of two prefactorization algebras is given by taking the tensor product of values at each open: (F ⊗ G)(U) = F(U) ⊗ G(U), and we simply define the structure maps as the tensor product of the structure maps. If the tensor product commutes (separately in each variable) with colimits over simplicial diagrams (i.e., geometric realizations), then the category of factorization algebras inherits this symmetric monoidal structure structure. The key point is to ensure that F ⊗ G satisfies the descent axiom, which follows quickly from the hypothesis on the tensor product. As we will not use this symmetric monoidal structure on factorization algebras, we do not develop this observation.

6.2 Factorization Algebras in Quantum Field Theory

215

6.2 Factorization Algebras in Quantum Field Theory We have seen in Section 1.3 in Chapter 1 how prefactorization algebras appear naturally when one thinks about the structure of observables of a quantum field theory. It is natural to ask whether the local-to-global axiom that distinguishes factorization algebras from prefactorization algebras also has a quantum fieldtheoretic interpretation. The local-to-global axiom we posit states, roughly speaking, that all observables on an open set U ⊂ M can be built up as sums of observables supported on arbitrarily small open subsets of M. To be concrete, let us consider a Weiss cover Uε of M, built out of all open discs in M of radius smaller than ε. Applied to this Weiss cover, our local-to-global axiom states that any observable O ∈ Obs(U) can be written as a sum of observables of the form O1 O2 · · · Ok , where Oi ∈ Obs(Dδi (xi )) and x1 , . . . , xk ∈ M. By taking ε to be very small, we see that our local-to-global axiom implies that all observables can be written as sums of products of observables whose support arbitrarily close to being point-like in U. This assumption is physically reasonable: most of the observables (or operators) considered in quantum field theory textbooks are supported at points, so it might make sense to restrict attention to observables built from these. However, more global observables are also considered in the physics literature. For example, in a gauge theory, one might consider the observable which measures the monodromy of a connection around some loop in the space–time manifold. How would such observables fit into the factorization algebra picture? The answer reveals a key limitation of our axioms: the concept of factorization algebra is appropriate only for perturbative quantum field theories. Indeed, in a perturbative gauge theory, the gauge field (i.e., the connection) is taken to be an infinitesimally small perturbation A0 + δA of a fixed connection A0 , which is a solution to the equations of motion. There is a well-known formula (the time-ordered exponential) expressing the holonomy of A0 + δA as a power series in δA, where the coefficients of the power series are given as integrals over Lk , where L is the loop that we are considering. This expression shows that the holonomy of A0 +δA can be built up from observables supported at points (which happen to lie on the loop L). Thus, the holonomy observable will form part of our factorization algebra. However, if we are not working in a perturbative setting, this formula does not apply, and we would not expect (in general) that the prefactorization algebra of observables satisfies the local-to-global axiom.

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6.3 Variant Definitions of Factorization Algebras In this section we sketch variations on the definition of (pre)factorization algebra used in this book. Our primary goal is to provide a bridge to the next section, where we discuss locally constant factorization algebras, which are well developed in the topology literature using an approach along the lines sketched here. The variations we describe undoubtedly admit improvements and modifications; they should not be taken as definitive. We hope and expect that approaches with this flavor may nonetheless play a role in future work. The first observation is that many constructions of (pre)factorization algebras make sense on a large class of manifolds and not just on a fixed manifold. As an example, consider the factorization envelope of a dg Lie algebra g. Under a smooth open embedding f : M → N, the natural map f∗ : ∗c (M) → ∗c (N) of extending by zero preserves the wedge product. Hence, given a dg Lie algebra g, we obtain a natural map f∗ ⊗ idg : ∗c (M) ⊗ g → ∗c (N) ⊗ g of dg Lie algebras. Thus, the pushforward f∗ Ug of the factorization envelope on M is isomorphic to the factorization envelope on N, when restricted to the image of M. In other words, this factorization envelope construction naturally lives on a category of n-manifolds, not just on a fixed n-manifold. We can formalize this kind of structure as follows. Definition 6.3.1 Let Embn denote the category whose objects are smooth nmanifolds and whose morphisms are open embeddings. It possesses a symmetric monoidal structure under disjoint union. Then we introduce the following variant of the notion of a prefactorization algebra. (In Section 6.3.4, we explain the appropriate local-to-global axiom.) Definition 6.3.2 A prefactorization algebra on n-manifolds with values in a symmetric monoidal category C is a symmetric monoidal functor from Embn to C. Observe the following property of such a functor F: the space of embeddings Embn (M, M) acts on the value F(M). In particular, the diffeomorphism group of M is a subset of Embn (M, M), and so F(M) has an action of the diffeomorphism group. Thus a prefactorization algebra F is sensitive to the topology of smooth manifolds; it is not a trivial generalization of the notion we’ve already developed. As an example, for any dg Lie algebra g, the factorization envelope Un g : M → C∗ ( ∗c (M) ⊗ g) provides a prefactorization algebra on n-manifolds.

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6.3.1 Varying in Smooth Families It would be nice, for instance, to let the diffeomorphism group act smoothly, in some sense, and not just as a discrete group. Within the framework we’ve developed, there is a natural approach, parallel to our definition of a smoothly equivariant prefactorization algebra (see Section 3.7.3 in Chapter 3). Note that embeddings define a natural sheaf of sets on the site Mfld of smooth manifolds: given n-manifolds M, N, the sheaf Emb(M, N) assigns to the manifold X the subset {f ∈ C∞ (X × M, N) | ∀x ∈ X, f (x, −) is an embedding of M in N}. In other words, it gives smooth families over X of embeddings of M into N. Definition 6.3.3 Let Embn denote the category enriched over Sh (Mfld) whose objects are smooth n-manifolds and in which Embn (M, N) is the sheaf Emb(M, N). It possesses a symmetric monoidal structure under disjoint union. If we work with a multicategory enriched over Sh(Mfld), such as differentiable vector spaces DVS, we can define a smooth prefactorization algebra as an enriched multifunctor (or map of enriched colored operads). This definition captures a sense in which we have smooth families of structure maps. Our primary example – the factorization envelope – provides such a smooth prefactorization algebra because ∗ (M) on a fixed manifold M defines a differentiable cochain complex.

6.3.2 Other Kinds of Geometry This kind of construction works very generally. For instance, if we are interested in complex manifolds, we could work in the following setting. Definition 6.3.4 Let Holn denote the category whose objects are complex nmanifolds and whose morphisms are open holomorphic embeddings. It possesses a symmetric monoidal structure under disjoint union. Definition 6.3.5 A prefactorization algebra on complex n-manifolds with values in a symmetric monoidal category C is a symmetric monoidal functor from Holn to C. Again, there is a natural example arising from a factorization envelope construction. Compactly supported Dolbeault forms 0,∗ c provide a functor from Holn to nonunital commutative dg algebras. Hence, tensoring with a dg Lie algebra g provides a symmetric monoidal functor from Holn to the category

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of dg Lie algebras, equipped with direct sum of underlying cochain complexes as a symmetric monoidal structure. Thus, the factorization envelope 0,∗ Uhol n g : M  → C∗ ( c (M) ⊗ g) provides a prefactorization algebra on complex n-manifolds. The Kac–Moody factorization algebras (see the example in Section 3.6.3 in Chapter 3) provide an example on Hol1 (i.e., for Riemann surfaces). In Williams (2016), Williams constructs the Virasoro factorization algebras on Hol1 . These examples extend naturally to Holn for any n, although the choices of central extension become more interesting and complicated. (Indeed, one finds extensions as L∞ algebras and not just dg Lie algebras.) It should be clear that one can enrich this category Holn over Sh(Mfld) and hence to talk about smooth families of such holomorphic embeddings. (Or, indeed, one can enrich it over a site of all complex manifolds.) Thus, one can talk, for instance, about smooth prefactorization algebras on complex nmanifolds. In general, let G denote some kind of local structure for n-manifolds, such a Riemannian metric or complex structure or orientation. In other words, G is a sheaf on Embn . A G-structure on an n-manifold M is a section G ∈ G(M). There is a category EmbG whose objects are n-manifolds with G-structure (M, GM ) and whose morphisms are G-structure-preserving embeddings, i.e., embeddings f : M → N such that f ∗ GN = GM . This category is fibered over EmbG . One can then talk about prefactorization algebras on G-manifolds.

6.3.3 Application to Field Theory This notion is quite useful in the context of field theory. One often studies a field theory that makes sense on a large class of manifolds. Indeed, physicists often search for action functionals that depend only on a particular geometric structure. For instance, a conformal field theory only depends on a conformal class of metric, and so the solutions to the equations of motion form a sheaf on the site of conformal manifolds. For example, the free βγ system makes sense on all conformal 2-manifolds, indeed, its solutions form a sheaf on Hol1 . Similarly, the physicist’s examples of a topological field theory depend only on the underlying smooth manifold (at least at the classical level). The solutions to the equations of motion for Abelian Chern–Simons theory (see Section 4.5 in Chapter 4) make sense on the site of oriented 3-manifolds. (More generally, Chern–Simons theory for gauge group G makes sense on the site of oriented 3manifolds with a principal G-bundle.) Finally, the free scalar field theory with mass m makes sense on the site of Riemannian n-manifolds.

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Since most of the constructions with field theories in this book have exploited the enveloping factorization algebra construction, the classical and quantum observables we have constructed are typically prefactorization algebras on Gmanifolds for some G. This universality of these quantizations – that these prefactorization algebras of quantum observables simultaneously work for all manifolds with some geometric structure – illuminates why our constructions typically recover standard answers: these are the answers that work generally. For interacting theories, the focus of Volume 2, the quantizations are rarely so easily obtained or described. On a fixed manifold, a quantization of the prefactorization algebra of classical observables might exist, but it might not exist for all manifolds with that structure. More explicitly, quantization proceeds via Feynman diagrammatics and renormalization, and hence it involves some explicit analysis (e.g., the introduction of counterterms). It is often difficult to quantize in a way that varies nicely over the collection of all G-manifolds. There are situations in which quantizations exist over some category of Gmanifolds, however. For instance, Li and Li (2016) construct a version of the topological B-model in this formalism and show that a quantization exists for all smooth oriented 2-manifolds after choosing a holomorphic volume form on the target complex manifold X.

6.3.4 Descent Axioms So far we have discussed only a variant of the notion of prefactorization algebra. We now turn to the local-to-global axiom in this context. Definition 6.3.6 A Weiss cover of a G-manifold M is a collection of G-embeddings {φi : Ui → M}i∈I such that for any finite set of points x1 , . . . , xn ∈ M, there is some i such that {x1 , . . . , xn } ⊂ φi (Ui ). With this definition in hand, we can formulate the natural generalization of our earlier definition. Definition 6.3.7 A factorization algebra on G-manifolds is a symmetric monoidal functor F : EmbG → Ch(DVS) that is a homotopy cosheaf in the Weiss topology. Let’s return to our running example. Our arguments in Section 6.6 show that the factorization envelope of a dg Lie algebra using the de Rham complex is a factorization algebra on Embn , and similarly the factorization enveloped using the Dolbeault complex is a factorization algebra on Holn . Remark: These definitions, indeed the spirit of this variation on factorization algebras, are inspired by conversations with and the work of John Francis and

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David Ayala. See Ayala and Francis (2015) for a wonderful overview of factorization algebras in the setting of topological manifolds, along with deep theorems about them. We note that they use the term “factorization homology,” rather than factorization algebra, as they view the construction as a symmetric monoidal analog, for n-manifolds, of ordinary homology. In this picture, the value on a disc defines the coefficients for a homology theory on n-manifolds. (We discuss factorization homology in the next section.) In Ayala et al. (2016), they have developed a generalization of factorization techniques where the coefficients are (∞, n)-categories. One payoff of their results is a proof of the Cobordism Hypothesis, and hence an explanation of functorial topological field theories via factorization-theoretic thinking. (See Lurie (2009c) for the canonical reference on the Cobordism Hypothesis.) ♦ Remark: The quantizations mentioned in the preceding text produce factorization algebras and not just prefactorization algebras. Hence, their values on big manifolds can be computed from their values on smaller manifolds, via the local-to-global axiom. It is interesting to compare this approach to understanding the global behavior of a quantum field theory – by a kind of open cover – with the approach in functorial field theory, in the style of Atiyah– Segal–Lurie, where a manifold is soldered together from manifolds with corners. The Ayala–Francis work shows that the factorization approach undergirds the cobordism approach, for topological field theories; the Cobordism Hypothesis is a consequence of their approach. Moving in a geometric direction, Dwyer, Stolz, and Teichner have suggested a method of using a factorization algebra on G-manifolds to build a functorial field theory on a category of G-cobordisms, appropriately understood. ♦

6.4 Locally Constant Factorization Algebras There is a family of operads that plays an important role in algebraic topology, known as the “little n-dimensional discs” operads, that were introduced by Boardman and Vogt (1973). The little n-discs operad is the operad in topological spaces whose k-ary operations is the space of embeddings of a disjoint union of k n-dimensional discs into a single n-dimensional disc via radial contraction and translation. (In other words, the embedding is completely specified by saying where the center of each disc goes and what its radius is.) We will use the term En operad for any topological operad weakly equivalent to the little n-discs, following a standard convention. An algebra over an En operad is called an En algebra. It has, by definition, families of k-fold multiplication

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maps parametrized by the configurations of k n-dimensional discs inside a single disc. If M is an n-dimensional manifold, then prefactorization algebras on M locally bear a strong resemblance to En algebras. After all, a prefactorization algebra prescribes a way to combine an element sitting on each of k distinct discs into an element for a big disc containing all k discs. In fact, En algebras form a full subcategory of factorization algebras on Rn , as we now indicate. Definition 6.4.1 A factorization algebra F on an n-manifold M is locally constant if for each inclusion of open discs D ⊂ D , then the map F(D) → F(D ) is a quasi-isomorphism. A central example is the locally constant factorization algebra Afact on R given by an associative algebra A. (Recall the example in Section 3.1.1.) Lurie has shown the following vast extension of this example. (See section 5.4.5 of Lurie (2016), particularly Theorem 5.4.5.9.) Theorem 6.4.2 There is an equivalence of (∞, 1)-categories between En algebras and locally constant factorization algebras on Rn . We remark that Lurie (and others) uses a different gluing axiom than we do. A careful comparison of the different axioms and a proof of their equivalence (for locally constant factorization algebras) can be found in Matsuoka (2014). We emphasize that higher categories are necessary for a rigorous development of these constructions and comparisons, although we will not emphasize that point in our discussion that follows, which is expository in nature. Remark: This theorem, together with our work in these two books, builds an explicit bridge between topology and physics. Topological field theories in the physicist’s sense, such as perturbative Chern–Simons theory, produce locally constant factorization algebras, which can then be viewed as En algebras and analyzed using the powerful machinery of modern homotopy theory. Conversely, intuition from physics about the behavior of topological field theories suggests novel constructions and examples to topology. Much of the amazing resonance between operads, homological algebra, and string-theoretic physics over the last few decades can be understood from this perspective. ♦

6.4.1 Motivation from Topology Many examples of En algebras arise naturally from topology, such as labelled configuration spaces, as discussed in the work of Segal (1973), McDuff (1975), B¨odigheimer (1987), Salvatore (2001), and Lurie (2016). We will discuss an important example, that of mapping spaces.

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Recall that (in the appropriate category of spaces) there is an isomorphism Maps(U V, X) ∼ = Maps(U, X)×Maps(V, X). This fact suggests that we might fix a target space X and define a prefactorization algebra by sending an open set U to Maps(U, X). This construction almost works, but it is not clear how to “extend” a map f : U → X from U to a larger open set V ⊃ U. By working with “compactly supported” maps, we solve this issue. Fix (X, p) a pointed space. Let F denote the prefactorization algebra on M sending an open set U to the space of compactly supported maps from U to (X, p). (Here, “f is compactly supported” means that the closure of f −1 (X − p) is compact.) Then F is a prefactorization algebra in the category of pointed spaces. (Composing with the singular chains functor gives a prefactorization algebra in chain complexes of Abelian groups, but we will work at the level of spaces.) Let’s consider for a moment the case when the open set is an open disc D ⊂ Rn = M. There is a weak homotopy equivalence F(D) " np X between the space of compactly supported maps D → X and the n-fold based loop space of X, based at the point p. (We are using the topologist’s notation np X for the n-fold loop space; hopefully, this use does not confuse due to the standard notation for the space of n-forms.) To see this equivalence, note that a compactly supported map f : D → X extends uniquely to a map from the closed D, sending the boundary ∂D to the base point p of X. Since np X is defined to be the space of maps of pairs (D, ∂D) → (X, p), we have constructed the desired map from F(D) to np X. It is easily verified that this map is a homotopy equivalence. Note that this prefactorization algebra is locally constant: if D → D is an inclusion of open discs, then the map F(D) → F(D ) is a weak homotopy equivalence. Note also that there is a natural isomorphism F(U1  U2 ) = F(U1 ) × F(U2 ) if U1 , U2 are disjoint opens. If D1 , D2 are disjoint discs contained in a disc D3 , the prefactorization structure gives us a map F(D1 ) × F(D2 ) → F(D3 ). These maps correspond to the standard En structure on the n-fold loop space. For a particularly nice example, consider the case where the source manifold is M = R. The structure maps of F then describe the standard product on the

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space p X of based loops in X. At the level of components, we recover the standard product on π0 p X = π1 (X, p). This prefactorization algebra F does not always satisfy the gluing axiom. However, Salvatore (2001) and Lurie (2016) have shown that if X is sufficiently connected, this prefactorization algebra is in fact a factorization algebra.

6.4.2 Relationship with Hochschild Homology The direct relationship between En algebras and locally constant factorization algebras on Rn raises the question of what the local-to-global axiom means from an algebraic point of view. Evaluating a locally constant factorization algebra on a manifold M is known as factorization homology or topological chiral homology. The following result, for n = 1, is striking and helpful. Theorem 6.4.3 For A an E1 algebra (e.g., an associative algebra), there is a weak equivalence Afact (S1 ) " HH∗ (A), where Afact denotes the locally constant factorization algebra on R associated to A and HH∗ (A) denotes the Hochschild homology of A. Here HH∗ (A) " A ⊗L A⊗A A means any cochain complex quasi-isomorphic to the usual bar complex (i.e., we are interested in more than the mere cohomology groups). In particular, this theorem says that the cohomology of the global sections on S1 of Afact equals the Hochschild homology of the E1 algebra A. This result is one of the primary motivations for the higher dimensional generalizations of factorization homology. It has several proofs in the literature, depending on choice of gluing axiom and level of generality (for instance, one can work with algebra objects in more general ∞-categories). See Theorem 3.9 of Ayala and Francis (2015) or Theorem 5.5.3.11 of Lurie (2016); a barehanded proof for dg algebras can be found in section 4.3 of Gwilliam (2012). There is a generalization of this result even in dimension 1. Note that there are more general ways to extend Afact from the real line to the circle, by allowing “monodromy.” Let σ denote an automorphism of A. Pick an orientation of S1 and fix a point p in S1 . Let (Afact )σ denote the prefactorization algebra on S1 such that on S1 − {p} it agrees with Afact but where the structure maps across p use the automorphism σ . For instance, if L is a small interval to the left of p, R is a small interval to the right of p, and M is an interval containing both L and

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R, then the structure map is A⊗A → a ⊗ b →

A a ⊗ σ (b)

where the leftmost copy of A corresponds to L and so on. It is natural to view the copy of A associated to an interval containing p as the A − A bimodule Aσ where A acts as the left by multiplication and the right by σ -twisted multiplication. Theorem 6.4.4 There is a weak equivalence (Afact )σ (S1 ) " HH∗ (A, Aσ ). Remark: In Section 8.1.2 in Chapter 8, we use this theorem to analyze the factorization homology of the quantum observables of the massive free scalar field on the circle. ♦ Beyond the one-dimensional setting, some of the most useful insights into the meaning of factorization homology arise from its connection to the cobordism hypothesis and fully extended topological field theories. See section 4 of Lurie (2009c) for an overview of these ideas, and Scheimbauer (n.d.) for development and detailed proofs. As mentioned in the remarks at the end of Section 6.3, Ayala et al. (2016) extends factorization homology by allowing coefficients in (∞, n)-categories, which leads to richer invariants. There is closely related work by Morrison and Walker (2012) using the formalism of blob homology.

6.4.3 Factorization Envelopes The connection between locally constant factorization algebras and En algebras also provides a beautiful universal property to the factorization envelope of a dg Lie algebra g, Un g : M → C∗ ( ∗c (M) ⊗ g), where M runs over the category Embn of smooth n-manifolds and embeddings. This enveloping construction provides a functor from dg Lie algebras to En algebras in the category of cochain complexes over a characteristic zero field. (Here we view Un g as an En algebra, by Lurie’s theorem.) On the other hand, there is a “forgetful” functor from this category of En algebras in cochain complexes to the category of dg Lie algebras. For n = 1, this functor is essentially the familiar functor from dg associative algebras to dg Lie algebras given by remembering only the commutator bracket and not the full multiplication. For n > 1, such a functor exists owing to the formality of the En operad: as

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an operad in cochain complexes, En " H ∗ (En ) = Pn , and there is clearly a forgetful functor from Pn algebras to dg Lie algebras. The factorization envelope provides the left adjoint for this forgetful functor: for any map of Lie algebras g → A, where A is an En algebra viewed as simply a Lie algebra, then there is a unique map Un g → A of En algebras, and conversely. To actually state this kind of result, we need to use ∞-categories, of course. Knudsen (2014) has provided a proof of this universal property using factorization techniques. Theorem 6.4.5 Let k be a field of characteristic zero. Let C be a stable, klinear, presentably symmetric monoidal ∞-category. There is an adjunction Un : AlgLie (C)  AlgEn (C) : F such that for any X ∈ C, FreeEn X " Un FreeLie ( n−1 X). Here FreeLie : C → AlgLie (C) denotes the free Lie algebra functor adjoint to the forgetful functor AlgLie (C) → C, and likewise FreeEn : C → AlgLie (C) denotes the free En algebra functor adjoint to the forgetful functor AlgEn (C) → C. As is conventional in topology, and hence in much work on stable ∞categories,  : C → C denotes the “suspension functor” sending X to the  pushout 0 X 0. In the setting of cochain complexes, suspension is given by a shift.

6.4.4 References For a deeper discussion of factorization homology than we’ve given, we recommend Ayala and Francis (2015) to start, as it combines a clear overview with a wealth of applications. A lovely expository account is Ginot (2015). Locally constant factorization algebras already possess a substantial literature, as they sit at the nexus of manifold topology and higher algebra. See, for instance, Ayala et al. (2015, 2016), Francis (2013), Ginot et al. (2012, 2014), Horel (2015), Lurie (2016), Matsuoka (2014), and Morrison and Walker (2012). Some striking applications to representation theory, particularly quantum character varieties, can be found in Ben Zvi et al. (2016).

6.5 Factorization Algebras from Cosheaves The goal of this section is to describe a natural class of factorization algebras. The factorization algebras that we construct from classical and quantum field theory will be closely related to the factorization algebras discussed here.

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The main result of this section is that, given a nice cosheaf of vector spaces or cochain complexes F on a manifold M, the functor Sym F : U → Sym(F(U)) is a factorization algebra. It is clear how this functor is a prefactorization algebra (see the example in Section 3.1.1 in Chapter 3); the hard part is verifying that it satisfies the local-to-global axiom. The examples in which we are ultimately interested arise from cosheaves F that are compactly supported sections of a vector bundle, so we focus on cosheaves of this form. We begin by providing the definitions necessary to state the main result of this section. We then state the main result and explain its role for the rest of the book. Finally, we prove the lemmas that culminate in the proof of the main result.

6.5.1 Preliminary Definitions Definition 6.5.1 A local cochain complex on M is a graded vector bundle E on M (with finite rank), whose smooth sections will be denoted by E , equipped with a differential operator d : E → E of cohomological degree 1 satisfying d2 = 0. Recall the notation from Section 3.5 in Chapter 3. For E be a local cochain complex on M and U an open subset of M, we use E (U) to denote the cochain complex of smooth sections of E on U, and we use Ec (U) to denote the cochain complex of compactly supported sections of E on U. Similarly, let E (U) denote the distributional sections on U and let E c (U) denote the compactly supported distributional sections of E on U. In Appendix B, it is shown that these four cochain complexes are differentiable cochain complexes in a natural way. We use E! = E ⊗ DensM to denote the appropriate dual object. We give E! the differential that is the formal adjoint to d on E. Note that, ignoring the differential, E c (U) is the continuous dual to E ! (U) and that Ec (U) is the ! continuous dual to E (U). Lemma 6.5.2 A local cochain complex (E , d) is a sheaf of differentiable cochain complexes. In fact, it is also a homotopy sheaf. Similarly, compactly supported sections (Ec , d) is a cosheaf of differentiable cochain complexes, as well as a homotopy cosheaf. Proof We describe the cosheaf case as the sheaf case is parallel. In Section B.7.2 in Appendix B, we showed that precosheaf Eck given the degree k vector space is a cosheaf with values in DVS. Thus we know that

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Ec , without the differential, is a cosheaf of graded differentiable vector spaces. As colimits of cochain complexes (note: not homotopy hocolimits) are computed degreewise, we see that (Ec , Q) is a cosheaf of differentiable cochain complexes. Identical arguments apply to E c . The homotopy cosheaf condition is only marginally more difficult. Suppose ˇ we are interested in an open U and a cover U for U. Recall that the Cech complex is the totalization of a double complex; we will view d as the vertical ˇ differential and the Cech differential as horizontal, so that the double complex is concentrated in the left half plane. We will work with the “augmented” double complex in which Ec (U) is added as the degree 1 column. The contracting cochain homotopy constructed in Section B.7.2 in Appendix B applies to the cosheaf Eck of degree k elements. Hence this augmented double complex is acyclic in the horizontal direction. The usual “staircase” or “zigzag” then ensures that the whole double complex is quasi-isomorphic to the rightmost column. The factorization algebras we will discuss are constructed from the symmet! ! ric algebra on the vector spaces Ec (U) and E c (U). Note that, since E c (U) is 5 E !c (U) as the algebra of formal power series dual to E (U), we can view Sym on E (U). Thus, we often write

5 E !c (U) = O(E (U)), Sym because we view this algebra as the space of “functions on E (U).” Note that if U → V is an inclusion of open sets in M, then there are natural maps of commutative dg algebras Sym Ec (U) → Sym Ec (V), 5 Ec (V), 5 Ec (U) → Sym Sym Sym E c (U) → Sym E c (V), 5 E c (U) → Sym 5 E c (V). Sym Thus, each of these symmetric algebras forms a precosheaf of commutative algebras, and thus a prefactorization algebra. We denote these prefactorization algebras by Sym Ec and so on.

6.5.2 The Main Theorem The main result of this section is the following. Theorem 6.5.3 We have the following parallel results for vector bundles and local cochain complexes.

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(i) Let E be a vector bundle on M. Then (a) Sym Ec and Sym E c are strict (nonhomotopical) factorization algebras valued in the category of differentiable vector spaces, and 5 E c are strict (nonhomotopical) factorization alge5 Ec and Sym (b) Sym bras valued in the category of differentiable pro-vector spaces. (ii) Let E be a local cochain complex on M. Then (a) Sym Ec and Sym E c are homotopy factorization algebras valued in the category of differentiable cochain complexes, and 5 E c are homotopy factorization algebras valued 5 Ec and Sym (b) Sym in the category of differentiable pro-cochain complexes. Proof Let us first prove the strict version of the result. To start with, consider the case of Sym Ec . We need to verify the local-to-global axiom. Let U be an open set in M and let U = {Ui | i ∈ I} be a Weiss cover of U. We need to prove that Sym∗ Ec (U) is the cokernel of the map   Sym(Ec (Ui ∩ Uj )) → Sym(Ec (Ui )) i,j∈I

i∈I

that sends a section on Ui ∩ Uj to its extension to a section of Ui minus its extension to Uj . This map preserves the decomposition of Sym Ec (U) into symmetric powers: a section of Symk (Ec (Ui ∩ Uj )) extends to a section of Symk (Ec (Ui )). Thus, it suffices to show that  Symm Ec (U) = coker ⊕i,j∈I Symm (Ec (Ui ∩ Uj )) → ⊕i∈I Symm Ec (Ui ) , for all m. Now, observe that 

Ec (U)⊗π m = Ecm (U m ) where Ecm is the cosheaf on U m of compactly supported smooth sections of the vector bundle Em , which denotes the external product of E with itself m times. Thus it is enough to show that ⎞ ⎛     Ecm (Ui ∩ Uj )m → Ecm Uim ⎠ . Ecm (U m ) = coker ⎝ i,j∈I

i∈I

Our cover U is a Weiss cover. This means that, for every finite set of points x1 , . . . , xm ∈ U, we can find an open Ui in the cover U containing every xj .

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This implies that the subsets of U m of the form (Ui )m , where i ∈ I, cover U m . Further, (Ui )m ∩ (Uj )m = (Ui ∩ Uj )m . The desired isomorphism now follows from the fact that Ecm is a cosheaf on M m . The same argument applies to show that Sym Ec! is a factorization algebra. In the completed case, essentially the same argument applies, with the subtlety (see Section C.4 in Appendix C) that, when working with pro-cochain complexes, the direct sum is completed. For the homotopy case, the argument is similar. Let U = {Ui | i ∈ I} be a Weiss cover of an open subset U of M. Let F = Sym Ec denote the prefactorization algebras we are considering (the argument that follows will apply when we use the completed symmetric product or use E c instead of Ec ). We need to show that the map ˇ C(U, F) → F(U) ˇ is an equivalence, where the left-hand side is equipped with the standard Cech differential. Let F m (U) = Symm Ec . Both sides above split as a direct sum over m, and the map is compatible with this splitting. (If we use the completed symmetric product, this decomposition is as a product rather than a sum.) We thus need to show that the map   Symm Ec Ui1 ∩ · · · ∩ Uin [n − 1] → Symm (Ec (U)) i1 ,...,in

is a weak equivalence. For i ∈ I, we get an open subset Uim ⊂ U m . Since U is a Weiss cover of U, these open subsets form a cover of U m . Note that m  Uim1 ∩ · · · ∩ Uimn = Ui1 ∩ · · · ∩ Uin . Note that Ec (U)⊗m can be naturally identified with c (U m , Em ) (where the tensor product is the completed projective tensor product). ˇ Thus, to show that the Cech descent axiom holds, we need to verify that the map    c Uim1 ∩ · · · ∩ Uimn , Em [n − 1] → c (V m , Em ) i1 ,...,in ∈I

ˇ is a quasi-isomorphism. The left-hand side above is the Cech complex for the m m on V . Standard partition of cosheaf of compactly supported sections of E

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unity arguments show that this map is a weak equivalence. (See Section A.5.4 in Appendix A for discussion and references.)

6.6 Factorization Algebras from Local Lie Algebras We just showed that for a local cochain complex E, the prefactorization algebra 5 Ec is, in fact, a factorization algebra. What this construction says is that Sym the “functions on E ,” viewed as a space, satisfy a locality condition on the manifold M over which E lives. We can reconstruct functions on E (M) from knowing about functions on E with very small support on M. But E is a simple kind of space, as it is linear in nature. (We should remark that E is a simple kind of derived space because it is a cochain complex.) We now extend to a certain type of nonlinear situation. In Section 3.6 in Chapter 3, we introduced the notion of a local dg Lie algebra. Since a local cochain complex is an Abelian local dg Lie algebra, we might hope that the Chevalley–Eilenberg cochain complex C∗ L of a local Lie algebra L also forms a factorization algebra. As we explain in Volume 2, a local dg Lie algebra can be interpreted as a derived space that is nonlinear in nature. In this setting, the Chevalley–Eilenberg cochain complex are the “functions” on this space. Hence, if C∗ L is a factorization algebra, we would know that functions on this nonlinear space L can also be reconstructed from functions localized on the manifold M. In Section 3.6 in Chapter 3, we also constructed an important class of prefactorization algebras: the factorization envelope of a local dg Lie algebra. Again, in the preceding section, we showed Sym Ec is a factorization algebra, which we can view as the factorization envelope of the Abelian local Lie algebra E [−1]. Thus, we might expect that the factorization envelope of a local Lie algebra satisfies the local-to-global axiom. We will now demonstrate that both these Lie-theoretic constructions are factorization algebras. Theorem 6.6.1 Let L be a local dg Lie algebra on a manifold M. Then the prefactorization algebras UL :U → C∗ (Lc (U)) OL :U → C∗ (L(U)) are factorization algebras. Remark: The argument that follows applies, with very minor changes, to a ♦ local L∞ algebra, a modest generalization we introduce later.

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231

Proof The proof is a spectral sequence argument, and we will reuse this idea throughout the book (notably in proving that quantum observables form a factorization algebra). We start with the factorization envelope. Note that for any dg Lie algebra g, the Chevalley–Eilenberg chains C∗ g have a natural filtration F n = Sym≤n (g[1]) compatible with the differential. The first page of the associated spectral sequence is simply the cohomology of Sym(g[1]), where g is viewed as a cochain complex rather than a Lie algebra (i.e., we extend the differential on g[1] as a coderivation to the cocommutative coalgebra Sym(g[1])). ˇ Consider the Cech complex of UL with respect to some Weiss cover U for an open U in M. Applying the filtration above to each side of the map ˇ C(U, UL) → UL(U), we get a map of spectral sequences. On the first page, this map is a quasiisomorphism by Theorem 6.5.3. Hence the original map of filtered complexes is a quasi-isomorphism. Now we provide the analogous argument for OL. For any dg Lie algebra g, the Chevalley–Eilenberg cochains C∗ g have a natural filtration F n = 5 ≥n (g∨ [−1]) compatible with the differential. The first page of the associSym 5 ∨ [−1]), where we ated spectral sequence is simply the cohomology of Sym(g ∨ view g [−1] as a cochain complex and extend its differential as a derivation to the completed symmetric algebra. ˇ Consider the Cech complex of OL with respect to some Weiss cover U for an open U in M. Applying the filtration above to each side of the map ˇ C(U, OL) → OL(U), we get a map of spectral sequences. On the first page, this map is a quasiisomorphism by Theorem 6.5.3. Hence the original map of filtered complexes is a quasi-isomorphism.

7 Formal Aspects of Factorization Algebras

In this chapter, we describe some natural constructions with factorization algebras. For example, we show how factorization algebras, like sheaves, push forward along maps of spaces. We also describe ways in which a factorization algebra can be reconstructed from local data.

7.1 Pushing Forward Factorization Algebras A crucial feature of factorization algebras is that they push forward nicely. Let M and N be topological spaces admitting Weiss covers and let f : M → N be a continuous map. Given a Weiss cover U = {Uα } of an open U ⊂ N, let f −1 U = {f −1 Uα } denote the preimage cover of f −1 U ⊂ M. Observe that f −1 U is Weiss: given a finite collection of points {x1 , . . . , xn } in f −1 U, the image points {f (x1 ), . . . , f (xn )} are contained in some Uα in U and hence f −1 Uα contains the xj . Definition 7.1.1 Given a factorization algebra F on a space M and a continuous map f : M → N, the pushforward factorization algebra f∗ F on N is defined by f∗ F(U) = F(f −1 (U)). Note that for the map to a point f : M → pt, the pushforward factorization algebra f∗ F is simply the global sections of F. We also call this the factorization homology of F on M.

7.2 Extension from a Basis Let X be a topological space, and let U be a basis for X, that is closed under taking finite intersections. It is well-known that there is an equivalence of categories between sheaves on X and sheaves that are defined only for open sets 232

7.2 Extension from a Basis

233

in the basis U. In this section we prove a similar statement for factorization algebras. In this section, we require topological spaces to be Hausdorff.

7.2.1 Preliminary Definitions We begin with a paired set of definitions. Definition 7.2.1 A factorizing basis U = {Ui }i∈I for a space X is a basis for the (usual) topology of X with the following properties: (i) For every finite set {x1 , . . . , xn } ⊂ X, there exists i ∈ I such that {x1 , . . . , xn } ⊂ Ui . (ii) If Ui and Uj are disjoint, then Ui ∪ Uj is in U. (iii) Ui ∩ Uj ∈ U for every Ui and Uj in U. As an example, for X = Rn , consider the collection of all opens that are a disjoint union of finitely many convex open sets. Note that a factorizing basis provides a Weiss cover for every open U. Given any finite set {x1 , . . . , xn } ⊂ U, fix a collection of pairwise disjoint opens Vi ⊂ U such that xi ∈ Vi . (Here we use that X is Hausdorff.) Then there is an element of the basis Ui ⊂ Vi with xi ∈ Ui . The disjoint union U1  · · ·  Un ⊂ U is in U and contains {x1 , . . . , xn } ⊂ U. Thus, the collection of opens in U contained in U form a Weiss cover for U. Let C denote a symmetric monoidal Grothendieck Abelian category. Definition 7.2.2 Given a factorizing basis U, a U-prefactorization algebra consists of the following: (i) For every Ui ∈ U, an object F(Ui ) ∈ C. (ii) For every finite tuple of pairwise disjoint opens Ui0 , . . . , Uin all contained in Uj – with all these opens from U – a structure map F(Ui0 ) ⊗ · · · ⊗ F(Uin ) → F(Uj )). (iii) The structure maps are equivariant and associative. In other words, a U-prefactorization algebra F is like a prefactorization algebra, except that F(U) is defined only for sets U in U. A U-factorization algebra is a U-prefactorization algebra with the property that, for every U in U and every Weiss cover V of U consisting of open sets in U, the natural map " ˇ → F(U) C(V, F) −

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ˇ ˇ is a weak equivalence, where C(V, F) denotes the Cech complex described earlier (Section 6.1 in Chapter 6). Note that we have not required that F(U) ⊗ F(V) → F(U  V) is an equivalence, so that we are focused here on factorization algebras without the multiplicativity condition. The extension construction will rely on having a lax symmetric monoidal functor from simplicial cochain complexes to cochain complexes via an Eilenberg–Zilber-type shuffle map. That is, we want the classical formula to hold in this setting. (Those familiar with higher categories will recognize that a similar proof works with any symmetric monoidal ∞-category whose symmetric monoidal product preserves sifted colimits.)

7.2.2 The Statement In this section we show that any U-factorization algebra on X extends to a factorization algebra on X. This extension is unique up to quasi-isomorphism. For an open V ⊂ X, we use UV to denote the collection of all the opens in U contained inside V. Let F be a U-factorization algebra. Let us define a prefactorization algebra ext(F) on X by ˇ V , F), ext(F)(V) = C(U for each open V ⊂ X. Conversely, if G is a factorization algebra on X, let res(G) denote its restriction to U-factorization algebra. Our main result here is then the following. Proposition 7.2.3 For any U-factorization algebra F, its extension ext(F) is a factorization algebra on X such that res(ext(F)) is naturally isomorphic to F. Conversely, if G is a factorization algebra on X, then ext(res(G)) is naturally quasi-isomorphic to G. Thus, we will prove that there is a kind of equivalence of categories res

FA(X, C)

FA(U, C) x ext

between the category PreFA(XC) of factorization algebras on X and the category PreFA(U, C) of U-factorization algebras. The subtlety here is that ext (res(F)) will not be isomorphic to F but just weakly equivalent.

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235

Remark: With these definitions in hand, it should be clear why we can recover a factorization algebra on X from just a factorization algebra on a factorizing basis U. The first property of U ensures that the “atomic” structure maps factor through the basis. (Compare with how an ordinary cover means that the “atomic” restriction maps of a sheaf factor through the cover: every sufficiently small neighborhood of any point is contained in some open in the cover.) The second property ensures that we know how to multiply within the factorizing basis. In particular, we have the maps F(Ui0 ) ⊗ · · · ⊗ F(Uin ) → F(Ui0 ∪ · · · ∪ Uin ) and F(Ui0 ∪ · · · ∪ Uin ) → F(Uk ), for any opens in our basis. Associativity thus determines the multiplication map F(Ui0 ) ⊗ · · · ⊗ F(Uin ) → F(Uk ). In other words, any multiplication map factors through a unary structure map and a multiplication map arising from disjoint union. Finally, the third property ensures that we know F on the intersections that appear in the gluing condition. ♦

7.2.3 The Proof Let U be a factorizing basis and F a U-factorization algebra. We will construct a factorization algebra ext(F) on X in several stages: • We give the value of ext(F) on every open V in X. • We construct the structure maps and verify associativity and equivariance under permutation of labels. • We verify that ext(F) satisfies the gluing axiom. Finally, it will be manifest from our construction how to extend maps of Ufactorization algebras to maps of their extensions. We use the following notations. Given a simplicial cochain complex A• (so each An is a cochain complex), let C(A• ) denote the totalization of the double complex obtained by taking the unnormalized cochains. Let CN (A• ) denote the totalization of the double complex by taking the normalized cochains. Finally, let shAB : CN (A• ) ⊗ CN (B• ) → CN (A• ⊗ B• )

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denote the Eilenberg–Zilber shuffle map, which is a lax symmetric monoidal functor from simplicial cochain complexes to cochain complexes. See Theorem A.2.8 in Appendix A for some discussion. Extending Values For an open V ⊂ X, recall UV denotes the Weiss cover of V given by all the opens in U contained inside V. We then define ˇ V , F). ext(F)(V) := C(U This construction provides a precosheaf on X. Extending Structure Maps ˇ The Cech complex for the factorization gluing axiom arises as the normalized cochain complex of a simplicial cochain complex, as should be clear from its ˇ construction. We use C(V, F)• to denote this simplicial cochain complex, so that ˇ ˇ F)• ), C(V, F) = CN (C(V, with F a factorization algebra and V a Weiss cover. We construct the structure maps by using the simplicial cochain complexes ˇ C(V, F)• , as many properties are manifest at that level. For instance, the unary maps ext(F)(V) → ext(F)(W) arising from inclusions V → W are easy to understand: UV is a subset of UW and thus we get a map between every piece of the simplicial cochain complex. We now explain in detail the map mVV  : ext(F)(V) ⊗ ext(F)(V  ) → ext(F)(V ∪ V  ), where V ∩ V  = ∅. Note that knowing this map, we recover every other multiplication map ext(F )(V) ⊗ ext(F )(V )

ext(F )(W)

mVV

ext(F )(V ∪ V ) by postcomposing with a unary map. ˇ V , F)n is the direct sum of the complexes The n-simplices C(U F(Ui0 ∩ · · · ∩ Uin ), with each Uik in V. The n-simplices of the tensor product of simplicial cochain ˇ V  , F)• are precisely the levelwise tensor product ˇ V , F)• ⊗ C(U complexes C(U

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237

ˇ V , F)n ⊗ C(U ˇ V  , F)n , which breaks down into a direct sum of terms C(U F(Ui0 ∩ · · · ∩ Uin ) ⊗ F(Uj0 ∩ · · · ∩ Ujn ) with the Uik ’s in V and the Ujk ’s in V  . We need to define a map ˇ V , F)n ⊗ C(U ˇ V  , F)n → C(U ˇ V∪V  , F)n mVV  ,n : C(U for every n, and we will express it in terms of these direct summands. Now (Ui0 ∩ · · · ∩ Uin ) ∪ (Uj0 ∩ · · · ∩ Ujn ) = (Ui0 ∪ Uj0 ) ∩ · · · ∩ (Uin ∪ Ujn ). Thus we have a given map

 F(Ui0 ∩ · · · ∩ Uin ) ⊗ F(Uj0 ∩ · · · ∩ Ujn ) → F (Ui0 ∪ Uj0 ) ∩ · · · ∩ (Uin ∪ Ujn ) , because F is defined on a factorizing basis. The right-hand term is one of the ˇ V∪V  , F)n . Summing over the direct summands on direct summands for C(U the left, we obtain the desired levelwise map. Finally, the composition ˇ ∩ V, F)• ) ⊗ CN (C(U ˇ V  , F)• ) CN (C(U sh



 ˇ V  , F)• ˇ ∩ V, F)• ⊗ C(U CN C(U 

CN (mVV  ,• )

ˇ V∪V  , F)• ) CN (C(U gives us mVV  . A parallel argument works to construct the multiplication maps from n disjoint opens to a bigger open. The desired associativity and equivariance are clear at the level of the simˇ V , F), since they are inherited from F itself. plicial cochain complexes C(U Verifying Gluing We have constructed ext(F) as a prefactorization algebra, but it remains to verify that it is a factorization algebra. Thus, our goal is the following. Proposition 7.2.4 The extension ext(F) is a cosheaf with respect to the Weiss topology. In particular, for every open subset W and every Weiss cover W, the ˇ complex C(W, ext(F)) is quasi-isomorphic to ext(F)(W).

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As our gluing axiom is simply the axiom for cosheaf – but using a funny ˇ class of covers – the standard refinement arguments about Cech homology apply. We now spell this out. For W an open subset of X and W = {Wj }j∈J a Weiss cover of W, there are two associated covers that we will use: (a) UW = {Ui ⊂ W | i ∈ I} and (b) UW = {Ui | ∃j such that Ui ⊂ Wj }. The first is just the factorizing basis of W induced by U, but the second consists of the opens in U subordinate to the cover W. Both are Weiss covers of W. To prove the proposition, we break the argument into two steps and exploit the intermediary Weiss cover UW . Lemma 7.2.5 There is a natural quasi-isomorphism "

ˇ ˇ W , F), f : C(W, ext(F)) → C(U for every Weiss cover W of an open W. Proof This argument boils down to combinatorics with the covers. Some extra ˇ notation will clarify what’s going on. In the Cech complex for the cover W, for instance, we run over n + 1-fold intersections Wj0 ∩ · · · ∩ Wjn . We will denote this open by W)j , where )j = (j0 , . . . , jn ) ∈ J n+1 , with J the index set for W. Since we are in intersections for which all the indices are pairwise distinct, we n+1 denote this subset of J n+1 . let J5 First, we must exhibit the desired map f of cochain complexes. The source ˇ complex C(W, ext(F)) is constructed out of F’s behavior on the opens Ui . Explicitly, we have   ˇ W , F)[n]. ˇ C(U C(W, ext(F)) = )j n≥0 )j∈J5 n+1

Note that each term F(U)i ) appearing in this source complex appears only once ˇ W , F). Let f send each such term F(U)) to its unique in the target complex C(U i image in the target complex via the identity. This map f is a cochain map: it clearly respects the internal differential of each term F(U)i ), and it is compatiˇ ble with the Cech differential by construction. Second, we need to show f is a quasi-isomorphism. We will show this by imposing a filtration on the map and showing the induced spectral sequence is a quasi-isomorphism on the first page. ˇ W , F) by We filter the target complex C(U   ˇ W , F) := F(U)i )[k]. F n C(U k≤n )i∈I5 k+1

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239

ˇ Equip the source complex C(W, ext(F)) with a filtration by pulling this filtration back along f . In particular, for any U)i = Ui0 ∩ · · · ∩ Uin , the preimage under f consists of a direct sum over all tuples W)j = Wj0 ∩ · · · ∩ Wjm such that every Uik is a subset of W)j . Here, m can be any nonnegative integer (in particular, it can be bigger than n). Consider the associated graded complexes with respect to these filtrations. The source complex has ⎛ ⎞ ˇ W , F) = Gr C(U

  n≥0 )i∈I5 n+1

⎜ F(U)i )[n] ⊗ ⎜ ⎝



m+1 such that U ⊂W ∀k )j∈J ik )j

⎟ C[m]⎟ ⎠.

The rightmost term (after the tensor product) corresponds to the chain complex for a simplex – here, the simplex is infinite-dimensional – and hence is contractible. In consequence, the map of spectral sequences is a quasi-isomorphism.

ˇ We now wish to relate the Cech complex on the intermediary UW to that on UW . ˇ W , F) and C(U ˇ W , F) are quasi-isomorphic. Lemma 7.2.6 The complexes C(U Proof We will produce a proof " " ˇ W , F) ← ˇ W , extW (F)) → ˇ W , F). C(U C(U C(U

Recall that UW is a factorizing basis for W. Then F, restricted to W, is a UW -factorizing basis. Hence, for any V ∈ UW and any Weiss cover V ⊂ UW of V, we have " ˇ C(V, F) → F(V). For any V ∈ UW , let UW |V = {Ui ∈ U | Ui ⊂ V ∩ Wj for some j ∈ J}. Note that this is a Weiss cover for V. We define ˇ W |V , F). extW (F)(V) := C(U By construction, the natural map extW (F)(V) → F(V) is a quasi-isomorphism. Thus we have a quasi-isomorphism " ˇ W , F), ˇ W , extW (F)) → C(U C(U

(7.2.3.1)

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ˇ by using the natural map from line (7.2.3.1) on each open in the Cech complex for UW . The map " ˇ W , extW (F)) → ˇ W , F) C(U C(U

arises by mimicking the construction in the preceding lemma.

7.3 Pulling Back Along an Open Immersion Factorization algebras do not pull back along an arbitrary continuous map, at least not in a simple way. On the other hand, when U is an open subset of a space X, a factorization algebra F on X clearly restricts to a factorization algebra on U. We would like to generalize this observation to open immersions f : Y → X and to have a pullback operation producing a factorization algebra f ∗ F on Y from a factorization algebra on X. In particular, we would like to have pullbacks to covering spaces. For simplicity (and because it encompasses all the examples we care about), let Y and X be smooth manifolds of the same definition and let f : Y → X be a local diffeomorphism, i.e., for any y ∈ Y there is some open neighborhood Uy such that f |Uy is a diffeomorphism from Uy to f (Uy ). We also restrict to multiplicative factorization algebras, a restriction that will be visible in the construction of the pullback construction f ∗ . Equip X with a factorizing basis as follows. Fix a Riemannian metric on X and let C = {Ci }i∈I be the collection of all strongly convex open sets in X. (A set U ⊂ M is strongly convex if for any two points in U, the unique minimizing geodesic in M between the points is wholly contained in U and no other geodesic connecting the points in is U.) As every sufficiently small ball is strongly convex, C forms a basis for the ordinary topology of X. Note that the intersection of two strongly convex opens is another strongly convex open. Let C denote the Weiss cover generated by C, which is thus a factorizing basis for X. There is a closely associated basis on Y. Let C denote the collection of all connected opens U in Y such that f (U) is in C. Let C be the factorizing basis of Y generated by C . Given F a multiplicative factorization algebra on X, let FC denote the associated multiplicative factorization algebra on the basis C. We now define a multiplicative factorization algebra f ∗ FC on the basis C as follows. To each U ∈ C , let f ∗ FC (U) = F(f (U)). To a finite collection of disjoint opens

7.4 Descent Along a Torsor

241

V1 , . . . , Vn in C , we assign f ∗ FC (V1  · · ·  Vn ) = f ∗ FC (V1 ) ⊗ · · · ⊗ f ∗ FC (Vn ), the tensor product of the values on those components, by multiplicativity. For U in C (and thus convex) and V in C such that V ⊂ U, equip f ∗ FC with the obvious structure map f ∗ FC (V) = F(V) → F(U) = f ∗ FC (U). By multiplicativity, this determines the structure map for any inclusion in the factorizing basis C j Vj ⊂ i Ui , where each Ui ∈ C . This inclusion is the disjoint union over i of the inclusions Ui ∩ (j Vj ) ⊂ Ui , and we have already specified the structure map for each i. Let f ∗ F denote the factorization algebra on Y obtained by extending f ∗ FC from the basis C . Note that although we used a choice of Riemannian metric in making the construction, the factorization algebra f ∗ F is unique up to isomorphism. It is enough to exhibit a map locally and the pullbacks are locally isomorphic on Y.

7.4 Descent Along a Torsor Let G be a discrete group acting on a space X. Recall the definition of discretely G-equivariant prefactorization algebra (see Definition 3.7.1 in Chapter 3). Proposition 7.4.1 Let G be a discrete group acting properly discontinuously on X, so that X → X/G is a principal G-bundle. Then there is an equivalence of categories between G-equivariant multiplicative factorization algebras on X and multiplicative factorization algebras on X/G. Proof If F is a factorization algebra on X/G, then f ∗ F is a G-equivariant factorization algebra on X. Conversely, let F be a G-equivariant factorization algebra on F. Let Ucon be the open cover of X/G consisting of those connected sets where the Gbundle X → X/G admits a section. Let U denote the factorizing basis for X/G

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generated by Ucon . We will define a U-factorization algebra F G by defining F G (U) = F(σ (U)), where σ is any section of the G-bundle π −1 (U) → U. Because F is G-equivariant, F(σ (U)) is independent of the section σ chosen. Since U is a factorizing basis, F G extends canonically to a factorization algebra on X/G.

8 Factorization Algebras: Examples

8.1 Some Examples of Computations The examples at the heart of this book have an appealing aspect: for these examples, it is straightforward to compute global sections – the factorization homology – because we do not need to use the gluing axiom directly. Instead, we can use the theorems in the preceding sections to compute global sections via analysis. By analogy, consider how de Rham cohomology, which exploits partitions of unity and analysis, compares to the derived functor of global sections for the constant sheaf. In this section, we compute the global sections of several examples we’ve already studied in the preceding chapters.

8.1.1 Enveloping Algebras Let g be a Lie algebra and let gR denote the local Lie algebra ∗R ⊗g on the real line R. Recall Proposition 3.4.1, which showed that the factorization envelope UgR recovers the universal enveloping algebra Ug. This factorization algebra UgR is also defined on the circle. Proposition 8.1.1 There is a weak equivalence UgR (S1 ) " C∗ (g, Ugad ) " HH∗ (Ug), where in the middle we mean the Lie algebra homology complex for Ug as a g-module via the adjoint action v · a = va − av, where v ∈ g and a ∈ Ug and va denotes multiplication in Ug. 243

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The second equivalence is a standard fact about Hochschild homology (see, e.g., Loday (1998)). It arises by constructing the natural map from the Chevalley–Eilenberg complex to the Hochschild complex, which is a quasiisomorphism because the filtration (inherited from the tensor algebra) induces a map of spectral sequences that is an isomorphism on the E1 page. Note one interesting consequence of this theorem: the structure map for an interval mapping into the whole circle corresponds to the trace map Ug → Ug/[Ug, Ug]. Proof The wonderful fact here is that we do not need to pick a Weiss cover and ˇ work with the Cech complex. Instead, we simply need to examine the cochain ∗ complex C∗ ( (S1 ) ⊗ g). One approach is to use the natural spectral sequence arising from the filtration F k = Sym≤k . The E1 page is C∗ (g, Ug), and one must verify that there are no further pages. Alternatively, recall that the circle is formal, so ∗ (S1 ) is quasi-isomorphic to its cohomology H ∗ (S1 ) as a dg algebra. Thus, we get a homotopy equivalence of dg Lie algebras ∗ (S1 ) ⊗ g " g ⊕ g[−1], where, on the right, g acts by the (shifted) adjoint action on g[−1]. Now, C∗ (g ⊕ g[−1]) ∼ = C∗ (g, Sym g), where Sym g obtains a g-module structure from the Chevalley–Eilenberg homology complex. Direct computation verifies this action is precisely the adjoint action of g on Ug (we use, of course, that Ug and Sym g are isomorphic as vector spaces).

8.1.2 The Free Scalar Field in Dimension 1 Recall from Section 4.3 in Chapter 4 that the Weyl algebra is recovered from the factorization algebra of quantum observables Obsq for the free scalar field on R. We know that the global sections of Obsq on a circle should thus have some relationship with the Hochschild homology of the Weyl algebra, but we will see that the relationship depends on the ratio of the mass to the radius of the circle. For simplicity, we restrict attention to the case where S1 has a rotationinvariant metric, radius r, and total length L = 2π r. Let SL1 denote this circle R/ZL. We also work with C-linear observables, rather than R-linear.

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Proposition 8.1.2 If the mass m = 0, then H k Obsq (SL1 ) is C[h] ¯ for k = 0, −1 and vanishes for all other k. If the mass m satisfies mL = in for some integer n, then H k Obsq (SL1 ) is C[h] ¯ for k = 0, −2 and vanishes for all other k. For all other values of mass m, H 0 Obsq (SL1 ) ∼ = C[h] ¯ and all other cohomology groups vanish. Hochschild homology with monodromy (recall Theorem 6.4.4) provides an explanation for this result. The equations of motion for the free scalar field locally have a two-dimensional space of solutions, but on a circle the space of solutions depends on the relationship between the mass and the length of the circle. In the massless case, a constant function is always a solution, no matter the length. In the massive case, there is either a two-dimensional space of solutions (for certain imaginary masses, because our conventions) or a zerodimensional space. Viewing the space of local solutions as a local system, we have monodromy around the circle, determined by the Hamiltonian. When the monodromy is trivial (so mL ∈ iZ), we are simply computing the Hochschild homology of the Weyl algebra. Otherwise, we have a nontrivial automorphism of the Weyl algebra. Note that this proposition demonstrates, in a very primitive example, that the factorization homology can detect spectral properties of the Hamiltonian. The proof will exhibit how the existence of solutions to the equations of motion (in this case, a very simple ordinary differential equation) affects the factorization homology. It would be interesting to develop further this kind of relationship. Proof We directly compute the global sections, in terms of the analysis of the local Lie algebra 

2 ∞ 1 +m ∞ 1 L = C (SL ) −−−→ C (SL )[−1] ⊕ Ch¯ where h¯ has cohomological degree 1 and the bracket is  0 1 [φ , φ ] = h¯ φ0φ1 SL1

with φ k a smooth function with cohomological degree k = 0 or 1. We need to compute C∗ (L). Fourier analysis allows us to make this computation easily. We know that the exponentials eikx/L , with k ∈ Z, form a topological basis for smooth functions on S1 and that   k2 2 ikx/L 2 ( + m )e = + m eikx/L . L2

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Hence, for instance, when m2 L2 is not a square integer, it is easy to verify that the complex +m2

C∞ (SL1 ) −−−→ C∞ (SL1 )[−1] has trivial cohomology. Thus, H ∗ L = Ch¯ and so H ∗ (C∗ L) = C[h], ¯ where in the Chevalley–Eilenberg complex h¯ is shifted into degree 0. When m = 0, we have H ∗ L = Rx ⊕ Rξ ⊕ Rh, ¯ where x represents the constant function in degree 0 and ξ represents the constant function in degree 1. The bracket on this Lie algebra is [x, ξ ] = h¯ L. Hence H 0 (C∗ L) ∼ = C[h] = C[h], ¯ and H −1 (C∗ L) ∼ ¯ and the remaining cohomology groups are zero. When m = in/L for some integer n, we see that the operator  + m2 has two-dimensional kernel and cokernel, both spanned by e±inx/L . By a parallel argument to the case with m = 0, we see H 0 (C∗ L) ∼ = C[h] = H −2 (C∗ L) and ¯ ∼ all other groups are zero.

8.1.3 The Kac–Moody Factorization Algebras Recall from the example in Section 3.6.3 in Chapter 3 and Section 5.5 in Chapter 5 that there is a factorization algebra on a Riemann surface  associated to every affine Kac–Moody Lie algebra. For g a simple Lie algebra with symmetric invariant pairing −, − g , we have a shifted central extension of the local Lie algebra 0,∗  ⊗ g with shift  α, ∂β g . ω(α, β) = 

Let Uω g denote the twisted factorization envelope for this extended local Lie algebra. Proposition 8.1.3 Let g denote the genus of a closed Riemann surface . Then H ∗ (Uσ g()) ∼ = H ∗ (g, Ug⊗g )[c] ω

where c denotes a parameter of cohomological degree 0.

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Proof We need to understand the Lie algebra homology C∗ ( 0,∗ () ⊗ g ⊕ Cc), whose differential has the form ∂ + dLie . (Note that in the Lie algebra, c has cohomological degree 1.) Consider the filtration F k = Sym≤k by polynomial degree. The purely analytic piece ∂ preserves polynomial degree while the Lie part dLie lowers polynomial degree by 1. Thus the E1 page of the associated spectral sequence has Sym(H ∗ (, O) ⊗ g ⊕ Cc) as its underlying graded vector space. The differential now depends purely on the Lie-theoretic aspects. Moreover, the differential on further pages of the spectral sequence are zero. In the untwisted case (where there is no extension), we find that the E1 page is precisely H ∗ (g, Ug⊗g ), by computations directly analogous to those for Proposition 8.1.1. In the twisted case, the only subtlety is to understand what happens to the extension. Note that H 0 ( 0,∗ () is spanned by the constant functions. The pairing ω vanishes if a constant function is an input, so we know that the central extension does not contribute to the differential on the E1 page. An alternative proof is to use the fact the 0,∗ () is homotopy equivalent to its cohomology as a dg algebra. Hence we can compute C∗ (H ∗ (, O)⊗g⊕Cc) instead.

We can use the ideas of Sections 4.6 and 4.7 in Chapter 4 to understand the “correlation functions” of this factorization algebra Uω g . More precisely, given a structure map Uω g (V1 ) ⊗ · · · ⊗ Uω g (Vn ) → Uω g () for some collection of disjoint opens V1 , . . . , Vn ⊂ , we will provide a method for describing the image of this structure map. It has the flavor of a Wick’s formula. Let ,O- denote the cohomology class of an element O of Uω g . We want to encode relations between cohomology classes. −1 As  is closed, Hodge theory lets us construct an operator ∂ , which vanishes on the harmonic functions and (0,1)-forms but provides an inverse to ∂ on the complementary spaces. (We must make a choice of Riemannian metric, of course, to do this.) Now consider an element a1 · · · ak of cohomological degree 0 in Uω g (), where each aj lives in 0,1 () ⊗ g ⊕ Cc.

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Then we have   −1   −1 −1 (∂ + dLie ) (∂ a1 )a2 · · · ak = ∂(∂ a1 )a2 · · · ak + dLie (∂ a1 )a2 · · · ak = ∂(∂ +

−1

a1 )a2 · · · ak

k  −1 [∂ a1 , aj ]a2 · · · aj · · · ak j=2

+

k  !  j=2



∂

−1

a1 , ∂aj

" g

c a2 · · · aj · · · ak .

When a1 is in the complementary space to the harmonic forms, we know −1 ∂∂ a1 = a1 so we have the relation 0 = ,a1 · · · ak - +

k  −1 ,[∂ a1 , aj ]a2 · · · aj · · · ak j=2

+

k   j=2



! −1 " ∂ a1 , ∂aj ,c a2 · · · aj · · · ak -, g

which allows us to iteratively reduce the “polynomial” degree of the original term ,a1 · · · ak - (from k to k − 1 or less, in this case). This relation looks somewhat complicated but it is easy to understand for k small. For instance, in k = 1, we see that 0 = ,a1 whenever a1 is in the space orthogonal to the harmonic forms. We get  ! " −1 −1 0 = ,a1 a2 - + ,[∂ a1 , a2 ]- + ∂ a1 , ∂a2 ,c

g

for k = 2.

8.1.4 The Free βγ System In Section 5.4 of Chapter 5 we studied the local structure of the free βγ theory; in other words, we carefully examined the simplest structure maps for the factorization algebra Obsq of quantum observables on the plane C. It is natural to ask about the global sections on a closed Riemann surface. There are many ways, however, to extend this theory to a Riemann surface . Let L be a holomorphic line bundle on . Then the L-twisted free βγ system has fields E = 0,∗ (, L) ⊕ 1,∗ (, L∨ ),

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where L∨ denotes the dual line bundle. The −1-symplectic pairing on fields is to apply pointwise the evaluation pairing between L and L∨ and then to integrate the resulting density. The differential is the ∂ operator for these holomorphic line bundles. q Let ObsL denote the factorization algebra of quantum observables for the L-twisted βγ system. Locally on , we know how to understand the structure maps: pick a trivialization of L and employ our work from Section 5.4 in Chapter 5. Proposition 8.1.4 Let bk = dim H k (, L). Then H ∗ ObsL () is a rank-one free module over C[h] ¯ and concentrated in degree −b0 − b1 . q

Proof As usual, we use the spectral sequence arising from the filtration F k = Sym≤k on Obsq . The first page just depends on the cohomology with respect to ∂, so the underlying graded vector space is   Sym H ∗ (, L)[1] ⊕ H ∗ (, L∨ ⊗ 1hol )[1] [h], ¯ where 1hol denotes the holomorphic cotangent bundle on . By Serre duality, we know this is the symmetric algebra on a +1-symplectic vector space, concentrated in degrees 0 and −1 and with dimension b0 + b1 in each degree. The remaining differential in the spectral sequence is the BV Laplacian for the pairing, and so the cohomology is spanned by the maximal purely odd element, which has degree −b0 − b1 .

8.2 Abelian Chern–Simons Theory and Quantum Groups In this section, we analyze Abelian Chern–Simons theory from a point of view complementary to our analysis in Section 4.5 in Chapter 4. It is standard in mathematics to interpret Chern–Simons theory in terms of a quantum group: line operators in Chern–Simons theory are representations of the quantum group, and expected values of line operators (which provide invariants of knots in 3-manifolds) can be constructed using the representation theory of the quantum group. In the language of physics, the connection between Chern0-Simons theory and the quantum group is explained as follows. In any topological field theory, one expects to be able to define a category whose objects are line operators and whose morphisms are interfaces between line operators. More explicitly, we have in mind fixing a line in space and every line operator in this category is supported on a line parallel to the fixed line. In a three-dimensional topological field theory (TFT), we thus have two directions orthogonal to the

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fixed direction, and we can imagine taking the operator product of line operators in this plane. This operator product expansion (OPE) is expected to give the category of line operators the structure of a braided monoidal category. For Chern–Simons theory, the expectation from theoretical physics is that the braided monoidal category of line operators is the category of representations of a quantum group. Our primary goal in this section is to implement this idea in the language of factorization algebras, but developed in the particular example of Abelian Chern–Simons theory. This example demonstrates, in a substantially simpler context, many of the ideas and techniques used to construct the Yangian from a non-Abelian four-dimensional gauge theory in Costello (2013b). (The gauge theory yielding the Yangian should be viewed as a variant of non-Abelian Chern–Simons theory that lives C × R2 . The theory is holomorphic in the C direction and topological along the R2 directions.) The construction here uses deep theorems about En algebras, Koszul duality, and Tannakian reconstruction, which take us outside the central focus of this text. So we will explain the structure of the argument and provide references to the relevant literature, but we will not give a complete proof. Interested readers encouraged to consult Costello (2013b) for more.

Overview of this Section Before we can explain the argument, we need to formulate the statement. Thus, q we begin by explaining how every En algebra – and in particular ObsCS , which is an E3 algebra – has an associated ∞-category of modules with a natural En−1 -monoidal structure. This terminology means that in this category of modules, there is a space of ways to tensor together modules, and this space is homotopy equivalent to the configuration space of points in Rn−1 . In our sitq uation, n = 3 so the ∞-category of modules for ObsCS will be E2 -monoidal, which is the ∞-categorical version of being braided monoidal. With these ideas about modules in hand, we can give a precise version of our claim and provide a strategy of proof. In particular, we explain how these modq ules for ObsCS are equivalent to the category of representations of a quantum group. We finish this section by connecting the concept of module for an E3 algebra to that of a line operator defined in a more physical sense. This will justify our q interpretation of the E2 -monoidal category of left modules of ObsCS as being the category of line operators for Abelian Chern–Simons theory. Remark: The kinds of assertions and constructions we want to make here rely on methods and terminology from higher category theory and homotopical

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algebra. Formalizing our ideas here would be a nontrivial endeavor, and orthogonal to our primary goals in this book. Hence we will use the language of higher categories here quite loosely, although our expressions can be made precise. In particular, in this section only, we will sometimes simply use “category” in place of ∞-category, except where the distinction becomes important. We will also use terminology like “functor” as if working with ordinary categories. ♦

8.2.1 Modules for En Algebras Even in classical algebra, there are different notions of module for an associative algebra: left modules, right modules, and bimodules. If we view associative algebras as one-dimensional in nature, it makes us suspect that for an n-dimensional algebra – such as an En algebra – there might be a proliferation of distinct notions reasonably called “modules,” and that suspicion is true. (See Ayala et al. (2015) for examples.) Nevertheless, in the context here, we will only need the notion of left (equivalently, right) modules. The reason is that every En algebra A has an “underlying” E1 algebra A, which is unique up to equivalence (we explain this “forgetful functor” momentarily). Thus we can associate to A its category of left modules, which we will denote LModA to emphasize that we mean left modules (and not bimodules). As we will explain, this category LModA is naturally equipped with an En−1 monoidal structure. More generally, every En algebra has an underlying Ek algebra for all 0 ≤ k ≤ n. This fact can be seen in two distinct ways: via the operadic definition or via the factorization algebra picture. They are both geometric in nature. We start with the operadic approach. Fix a linear inclusion of ι : Rk → Rn . We will use ι : (x1 , . . . , xk ) → (x1 , . . . , xk , 0, . . . , 0). Then if Dkr (p) denotes the k-disc of radius r in Rk centered at p, then Dnr (ι(p)) ∩ ι(Rk ) = ι(Dkr (p)), where Dnr (ι(p)) denotes the n-disc of radius r in Rn centered at p. In this way, ι induces a map ι(m) : Ek (m) → En (m) from the configuration space of m disjoint little k-discs to the configuration space of m disjoint little n-discs. Altogether, we get a map ι : Ek → En of topological operads. Algebras over operads pull back along maps of operads, so for A ∈ AlgEn , we obtain ι∗ A ∈ AlgEk . One can rephrase this from the point of view of factorization algebras. We will need a construction of factorization algebras that we have not used before. If M is a manifold, F a factorization algebra on M, and X ⊂ M a submanifold (possibly with boundary), we can define a factorization algebra i∗X F by setting i∗X F(U) = lim F(V) V⊃U

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where the limit is taken over those opens V in X which contain U. The factorization product on F defines one on i∗X F. If F is locally constant, this construction is particularly nice: in that case, the limit is eventually constant so that i∗X F(U) is F(V) for some sufficiently small open V containing U. If R ⊂ Rn is a line, we can restrict a locally constant factorization algebra on Rn to one on R in this way, to produce an associative algebra. This restriction procedure is how one describes the E1 algebra underlying an Ek algebra in the language of factorization algebras.

Modules via Geometry We now want to examine the type of monoidal structure on LModA arising from an En algebra structure on A. Again, one can use results about operads and categories or one can use factorization algebras. Here we begin with the factorization approach. Let Hn = {(x1 , . . . , xn ) ∈ Rn | xn ≥ 0} denote the “upper half space.” Let ∂Hn denote the wall {xn = 0}. By “disc in H” we will mean the intersection D ∩ Hn for any open disc D ⊂ Rn . Since we are always working in Hn for the moment, we will use an open U to denote its intersection with Hn . Let A be a locally constant factorization algebra on Rn associated to the En algebra A. Given M ∈ LModA and a point p ∈ ∂Hn living on the wall {xn = 0}, we construct a factorization algebra Mp on Hn by extending from the following factorization algebra on a basis. For an open U not containing p, we set Mp (U) = A(U), and for any disc D in Hn containing p, we set Mp (D) = M. For any disjoint union U  D with p ∈ D, we set Mp (U  D) = A(U) ⊗ M. The structure maps are easy to specify. If U  D ⊂ D , then U ⊂ D \ D, and on this hemispherical shell, A(D \ D) " A. Since we know how A acts on M, we have the composition Mp (U) ⊗ Mp (D) = A(U) ⊗ M → A(D \ D) ⊗ M → A ⊗ M → M. Now extend Mp from this basis to all opens. To help visualize this factorization algebra, here is a picture in dimension 2.

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U2 A⊗A⊗M U1

 M

V2

V1 p

To each of the opens U1 and U2 , Mp assigns a copy of A. To the half-discs V1 and V2 , Mp assigns the left A-module M. Remark: It can help to use the following pushforward to understand the structure maps. The distance-from-p function dp : x → d(p, x) sends Hn to R≥0 . Pushing forward Mp along dp produces the factorization algebra encoding A on R>0 with M supported at 0, as discussed in Section 3.3 in Chapter 3. This pushforward makes clear how the action on M of A, evaluated on complicated opens in Hn , factors through the action of A. This pushforward encodes precisely the action of A on M with which we began. ♦ The same type of procedure allows us to assign a factorization algebra behaving like A in the bulk but with modules “inserted” at points on the boundary. Let p1 , . . . , pk be distinct points on Hn . Let M1 , . . . , Mk be objects of LModA . Abusively, let p) denote this collection {(Mi , pi )}. We construct a factorization algebra Mp) as follows. For any open U disjoint from the points {pi }, set Mp) (U) = A(U). For any disc D containing exactly one distinguished point pi , set Mp) (D) = Mi . For any disc D containing U  D, with D and D containing pi but not other distinguished point pj , we provide the structure map as above. In other words, in a sufficiently small neighborhood of pi , Mp) becomes Mpi as described earlier. Extend to all opens to produce Mp) . In the picture that follows, we use Mp to denote the left A-module associated with the point p and Mq to denote the left A-module associated with the point q.

V3

A ⊗ Mp ⊗ Mq

U

V1

 Mp ⊗A Mq

V2 p

q

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To the half-disc V1 , Mp,q assigns Mp ; to the half-disc V2 , Mp,q assigns Mq ; and to the open U, Mpq assigns A. The value on the big half-disc V3 must receive maps from all these values separately and all intermediary opens (e.g., through a half-disc containing V2 and U but not V1 ), so its value must be some version of “Mp ⊗A Mq .” From such an Mp) , we can recover a left A-module in a very simple way. Consider the projection map π : Hn → R≥0 sending (x1 , . . . , xn ) to xn . Then π∗ Mp) returns A on any interval not containing the boundary point 0, but it has interesting value on an interval containing 0. In this way, we produce a left A-module for any configuration of distinct points {p1 , . . . , pk } in ∂Hn ∼ = Rn−1 and any corresponding k-tuple of left Amodules {M1 , . . . , Mk }. In other words, we have sketched a process that can lead to making LModA an En−1 algebra in some category of categories. Modules via Operads Developing the picture sketched earlier into a formal theorem would be somewhat nontrivial. Fortunately, Lurie has proved the theorem we need in Lurie (2016), using the operadic approach to thinking about En algebras. We begin by stating the relevant result from his work and then gloss the essential argument (although we hide a huge amount of technical machinery developed to make such an argument possible). We need some notation to state the result. If C is a symmetric monoidal ∞-category, then C is itself an E∞ algebra in the ∞-category Cat∞ of ∞categories, equipped with the Cartesian symmetric monoidal structure (i.e., the product of categories). Hence, it makes sense to talk about left module categories over C, which we denote LModC (Cat∞ ). Let Cat∞ ( ) denote the subcategory of ∞-categories possessing geometric realizations and whose functors preserve geometric realizations. Theorem 8.2.1 (Theorem 4.8.5.5 and Corollary 5.1.2.6 of Lurie (2016)) Let C denote a symmetric monoidal ∞-category possessing geometric realizations and whose tensor product preserves colimits separately in each variable. Then there is a fully faithful symmetric monoidal functor LMod− : AlgE1 (C) → LModC (Cat∞ ( )) sending an E1 algebra A to LModA . This functor determines a fully faithful functor AlgEn (C) → AlgEn−1 (LModC (Cat∞ ( ))).

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This result might seem quite technical but it is a natural generalization of more familiar constructions from ordinary algebra. One might interpret the first part as saying that (i) There is a big category whose objects are categories with an action of the symmetric monoidal category C. (ii) The category of left modules over an algebra in C gives an object in this big category. (iii) There is a way of “tensoring” categories of modules such that LModA  LModB " LModA⊗B . Tensoring categories might not be familiar, but it is suggested by generalizing the construction of the tensor product of ordinary vector spaces: given a product of categories D × D , we might hope there is a category D  D receiving a bifunctor from D×D such that all other bifunctors from D×D factor through it. Note that we have a natural bifunctor LModA × LModB → LModA⊗B . To understand the second part of the theorem, it helps to know about Dunn additivity: it is equivalent to give an object A ∈ C the structure of an En algebra as it is to give A first the structure of an E1 algebra and then the structure of an En−1 algebra in the category of E1 algebras. In other words, there is an equivalence of categories AlgEn−1 (AlgE1 (C)) " AlgEn (C). (See Theorem 5.1.2.2 of Lurie (2016).) By induction, to give A an En algebra is to specify n compatible E1 structures on A. Geometrically, this assertion should seem plausible. If one has an En structure on A, then each way of stacking discs along a coordinate axes in Rn provides an E1 multiplications. Hence, we obtain n different E1 multiplications on A. In the other direction, knowing how to multiply along these axes, one should be able to reconstruct a full En structure. The theorem then says that we get a natural functor LMod

AlgEn (C) " AlgEn−1 (AlgE1 (C)) −−−→ AlgEn−1 (LModC (Cat∞ ( ))), where the first equivalence is Dunn additivity and the second functor exists because LMod− is symmetric monoidal. This composed arrow is what allows one to turn an En algebra into an En−1 monoidal category.

8.2.2 The Main Statement and the Strategy of Proof Let ACS denote the E3 algebra associated to the locally constant factorization q algebra ObsCS of quantum observables for Abelian Chern–Simons theory with

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the rank 1 Abelian Lie algebra. Since we will do everything over the field of complex numbers, we will identify this Lie algebra with C. By the theory described earlier, the category LModACS is an E2 -monoidal category. Recall that E2 -monoidal categories are the ∞-categorical version of braided monoidal categories. There is another braided monoidal category intimately connected with Chern–Simons theory: representations of the quantum group. We want to relate these two categories. In the case of interest, the quantum group is a quantization of the universal enveloping algebra of the Abelian Lie algebra C. Typically, a quantum group depends on a parameter often denoted q or h. ¯ The size of this parameter is related to the choice of an invariant pairing on the Lie algebra. In the Abelian case, multiplication by scalars is a symmetry of the Lie algebra that also scales the invariant pairing. Thus, all values of the quantum parameter are equivalent, and we find only a single quantum group. It will be convenient to consider a completed version of the universal envelop(C) = C[[t]]. The ing algebra. Thus, our Abelian quantum group will be U Hopf algebra structure will be that on the universal enveloping algebra. Thus, it is commutative and co-commutative and the variable t is primitive. The only interesting structure we find is the R-matrix, given by the formula (C)⊗ (C). U R = et⊗t ∈ U This R-matrix gives us a quasi-triangular Hopf algebra (in a completed sense). It is a standard result in the theory of quantum groups that the category of modules over a quasi-triangular Hopf algebra is a braided monoidal category. (C)) denote the category of finite-dimensional representaWe will let Rep(U tions of our quantum group. Such a finite-dimensional representation is the same as a vector space V with a nilpotent endomorphism. (C)) with a braided monoidal category We would like to compare Rep(U arising from our E3 -algebra ACS . The category of left modules over this E3 algebra is an E2 -monoidal ∞-category. To construct a braided monoidal category, we need to turn this ∞-category into an ordinary category, and we need (C)). this ordinary category to be equivalent as a monoidal category to Rep(U Identifying the Category of Interest To isolate the classical category we need, the following technical proposition will be useful. Proposition 8.2.2 Let us view ACS as an E2 algebra via the map of operads from E2 to E3 . Then there is a canonical equivalence of E2 algebras from ACS

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to the commutative algebra C[ε], where the parameter ε is of degree 1. We view C[ε] as an E2 algebra via the map from E2 to E∞ . Since the proof is both technical and unilluminating, we will defer it to the end of this section. With this proposition in hand, we can understand the category of modules over ACS much better. Since ACS is equivalent as an E2 algebra to C[ε], the monoidal category LModACS is equivalent to the monoidal category LModC[ε] . We will isolate in this monoidal ∞-category, a subcategory whose homotopy (C)). category is equivalent to Rep(U Let us start by indicating why there is any reason to expect a relationship between such categories. A first observation is that Koszul duality of associative algebras interchanges exterior and symmetric algebras. Indeed, C[ε] is an exterior algebra on one variable, and C[[t]] is a (completed) symmetric algebra on one variable, and they are Koszul dual to one another. Alternatively, C[[t]] is the (completed) universal enveloping algebra of the abelian Lie algebra C, C[ε] is the Chevalley–Eilenberg cochains of the same Lie algebra, and Koszul duality interchanges the universal enveloping algebras and the Chevalley–Eilenberg cochains of a given Lie algebra. It is then well known that Koszul duality induces an equivalence between (appropriate) categories of modules. In the case at hand, this relationship works (C), in which the as follows. If V is a finite-dimensional representation of U generator t acts by a nilpotent endomorphism ρ(t) of V, we define a dg module (V) for C[ε] by setting (V) = V ⊗ C[ε], d = ερ(t). More formally, since V is a module for the Lie algebra C, we can take the Chevalley–Eilenberg cochains C∗ (C, V) of C with coefficients in V. This complex is a dg module for C∗ (C) = C[ε]. Let LMod0 (C[ε]) be the full subcategory of the ∞-category of C[ε]-modules given by the essential image of the functor . One can describe this category in a different way. There is a functor Aug :

LMod(C[ε]) → M →

dgVect M ⊗L C[ε] C.

We call this functor Aug because it is constructed from the augmentation map C[ε] → C sending ε → 0. Notice that Aug is a monoidal functor, since the map C[ε] → C is a map of E2 algebras. The category LMod0 (C[ε]) then consists of those modules M for which Aug(M) has finite-dimensional cohomology concentrated in degree 0. (In other words, it is the homotopy fiber

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product of Aug and the inclusion of finite-dimensional vector space Vectfd into dgVect.) Since, as we have seen, ACS is equivalent to C[ε] as an E2 algebra, and so also as an E1 algebra, we can view this category as being a subcategory of LMod(ACS ). The Main Result (C)) is the homotopy The classical category we want to compare with Rep(U 0 0 category Ho(LMod (ACS )). Note that, since LMod (ACS ) is an E2 -monoidal category, its homotopy category is a braided monoidal category. Our main theorem is the following. (C)) with the structure of braided monoidal catTheorem 8.2.3 Equip Rep(U t⊗t egory by the R-matrix R = e for the Abelian quantum group. The functor (C)) → Ho(LMod0 (ACS ))  : Rep(U is an equivalence of braided monoidal categories, where The interesting part of this theorem is that it replaces the somewhat mysterious E3 algebra ACS with the much more explicit and concrete quasi-triangular Hopf algebra C[[t]] with R-matrix et⊗t . The Argument We will break the proof up into several lemmas. Lemma 8.2.4 The functor (C)) → Ho(LMod0 (ACS ))  : Rep(U is an equivalence of monoidal categories. This equivalence is as ordinary categories, not ∞-categories. Proof Since ACS is equivalent as an E2 algebra to C[ε], the monoidal product on the category of C[ε]-modules is given by tensoring over C[ε]. Let us check (C) = C[[t]], on which t that  is monoidal. If V, W are representations of U acts by ρV (t), ρW (t), respectively, then t acts on V ⊗W by ρV (t)⊗1+1⊗ρW (t). There is an isomorphism (V)⊗C[ε] (W) = (C[ε]⊗V)⊗C[ε] (C[ε]⊗W) → C[ε]⊗(V ⊗W) = (V ⊗W) that takes the differential on the left-hand side, coming from the tensor product of the differentials on (V) and (W), to the differential on (V ⊗ W). This natural isomorphism makes  monoidal. Next, let us check that  is an

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equivalence of categories. By the definition of LMod0 (ACS ), the functor  is essentially surjective. It is easy to verify that  is full and faithful. Lemma 8.2.5 The equivalence of the previous lemma transfers the braided monoidal category structure on Ho(LMod0 (ACS )) to a braided monoidal cat(C)). This braided monoidal category structure is realized by egory on Rep(U some R-matrix (C)⊗ (C), U R∈U (C) the structure of quasi-triangular Hopf algebra. giving U Proof Let us sketch how one can see this using the Tannakian formalism, before giving a more concrete approach to constructing the quasi-triangular Hopf algebra structure. The category Ho(LMod0 (ACS ) is a braided monoidal category. There is a functor Aug : Ho(LMod0 (ACS )) → Vect sending a module M to M ⊗C[ε] C, using the equivalence of E2 algebras between ACS and C[ε]. The augmentation map C[ε] → C is a map of E2 algebras, so this augmentation functor is monoidal. Under the equivalence (C)) " Ho(LMod0 (ACS )) Rep(U the functor Aug becomes the forgetful functor, which takes a representation of (C) and forgets the U (C) action. U Let us recall how the Tannakian formalism works for braided monoidal categories. (See, e.g., Etingof et al. (2015).) Suppose we have a braided monoidal category C with a monoidal (but not necessarily braided monoidal) functor F : C → Vect. Then, by considering the automorphisms of the fibre functor F, one can construct a quasi-triangular Hopf algebra whose representations will be our original braided monoidal category C, and where the functor F is the forgetful functor. (One needs some additional technical hypothesis on C and F for this construction to work.) We are in precisely the situation where this formalism applies. The braided monoidal category is Ho(LMod0 (ACS ), and the fiber monoidal functor is the functor Aug given by the augmentation of ACS as an E2 algebra. Since we already know that Ho(LMod0 (ACS )) is equivalent to representations of the (C), in such a way that the fiber functor becomes the forgetHopf algebra U (C). We ful functor, the Hopf algebra produced by the Tannakian story is U therefore find that the braiding is described by some R-matrix on this Hopf algebra.

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Since this approach is so abstract, let us describe how to see concretely the (C). R-matrix on U The equivalence with Ho(LMod0 (ACS )) gives us a braided monoidal struc(C). For any two representations V, W of U (C), ture on representations of U ∼ this braiding gives an isomorphism V ⊗ W = W ⊗ V. There is already such an isomorphism, just of vector spaces, since vector spaces are a symmetric monoidal category. Composing these two isomorphisms gives, for all repre∼ = (C), an isomorphism RV,W : V ⊗ W − → V ⊗ W. It is an sentations V, W of U (C)-modules, and it is isomorphism of vector spaces and not necessarily of U natural in V and W. The functor (C)) × Rep(U (C)) Rep(U (V, W)

→ →

Vect V ⊗W

2 (C)⊗ (C), i.e., the functor is (C)⊗ U module U is represented by the right U given by tensoring with this right module. The natural automorphism RV,W of this functor is thus represented by an automorphism of this right module, which must be given by left multiplication with some element

(C)⊗ (C). U R∈U This element is the universal R-matrix. Now, the axioms of a braided monoidal category translate directly into the properties that R must satisfy to define a quasi-triangular Hopf algebra struc(C). ture on U

So far, we have seen that we can identify the braided monoidal category of (C) with the braided monoidal category Ho LMod0 (ACS ), representations of U  where we equip U (C) with some R-matrix. It remains to show that there is essentially only one possible R-matrix. (C) satisfying the axioms defining a Proposition 8.2.6 Every R-matrix for U quasi-triangular Hopf algebra is of the form (C)⊗ (C) U R = exp(c(t ⊗ t)) ∈ U where c is a complex number.

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Proof Recall that an R-matrix of a Hopf algebra A is an invertible element R of A ⊗ A such that

op (x) = R (x)R−1 for all x ∈ A,

⊗ idA (R) = R13 R23 , idA ⊗ (R) = R13 R12 , where denotes the coproduct on A, op denotes postcomposed with the flip operation, and R13 , for example, denotes  ai ⊗ 1 ⊗ bi ∈ A ⊗ A ⊗ A i



where R = i ai ⊗ bi ∈ A ⊗ A. For us, owing to the completion of the tensor product, an element R will live C[[t2 ]]. As C[[t]] is cocommutative, we see that the first equation in C[[t1 ]]⊗ will be trivially satisfied by any invertible element. The only conditions are then the remaining equations. As we are working with a deformation of the usual symmetric braiding on C[[t]]-modules, an R-matrix is of the form 1 ⊗ 1 + c1 t1 ⊗ 1 + c2 1 ⊗ t2 + · · · . To simplify the problem further, note that, because C[[t1 , t2 ]] is a commutative algebra, we can take the logarithm of R. If we let R denote this logarithm, then the equations then become additive:

⊗ idA (R) = R13 + R23 , idA ⊗ (R) = R13 + R12 .  Writing R = i x(i) ⊗ y(i) with x(i) ∈ C[[t1 ]] and y(i) ∈ C[[t2 ]], we see that the first equation becomes  

(x(i) ) ⊗ y(i) = (x(i) ⊗ 1 + 1 ⊗ x(i) ) ⊗ y(i) , i

i

which forces each x(i) to be primitive. The other equation likewise forces each y(i) to be primitive. Hence R = ct1 ⊗ t2 for some constant c and so R = exp(ct1 ⊗ t2 ). The constant c is the only freedom available in choosing an R-matrix. Thus, we have shown the desired equivalence of braided monoidal cate(C) is equipped with an R-matrix of the form exp(c(t ⊗ t)) gories, where U

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for some value of the constant c. This constant cannot be zero, as this would force ACS to be a commutative E3 algebra, and ACS is not commutative. Any two nonzero values of the constant c are related by scaling t, and so all the braided monoidal categories are equivalent. This completes the proof of Theorem 8.2.3.

8.2.3 Line Operators In this section, we explain a general definition of a Wilson line operator in the language of factorization algebras, and we will attempt to match our definition with ideas prevalent in the physics literature. We then show that, in the case of Abelian Chern–Simons theory, the category of Wilson line operators is the category we considered earlier. The Basic Idea of Defects Let us first explain how line operators (and more general defects) are understood from the physics point of view. Suppose we have a field theory on a manifold M, and X ⊂ M is a submanifold. A defect for our field theory on X is a way of changing the field theory on X while leaving it the same outside X. Often, one can realize a defect by coupling a theory living on X to our original theory living on M. In the case that X is a one-dimensional submanifold, we call such a defect a line operator (or line defect). For example, if K : S1 → R3 is a knot, we could couple a free fermion on 1 S to Abelian Chern–Simons theory on R3 . If A is the field of Chern–Simons theory and ψ : S1 → C the fermionic field on S1 , we can couple the theories with the action   ∗ ψ(θ )(d + K A)ψ(θ ) + A ∧ dA. SK (A, ψ) = S1

R3

The quantization of this field theory, which we will not develop here, should q q produce a factorization algebra of observables ObsCS,K that agrees with ObsCS on opens disjoint from K but that on opens intersecting K also includes observables on ψ. A small tubular neighborhood  K of K inside R3 is diffeomorphic q 2 to K × R , so we can pushforward ObsCS,K from this neighborhood to S1 . The underlying E1 algebra AK should look something like ACS tensored with a Clifford algebra (the observables for the uncoupled free fermion). Thus the line defect for Abelian Chern–Simons theory produces an interesting modifiq q cation ObsCS,K of ObsCS near K; we will view this factorization algebra as a model for what a line defect should be in the setting of factorization algebras.

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By picking different field theories supported on K and coupling them to Chern–Simons theory in the bulk, we obtain other factorization algebras. One can do this, for instance, by varying the coupling constants in the example above or adding more interactions terms. One can similarly imagine picking a two-dimensional submanifold and coupling a field theory on it to the threedimensional bulk. Defects via Factorization Algebras Let us now explain in the language of factorization algebras how to couple a field theory to a quantum mechanical system living on a line. Suppose we have a factorization algebra F on Rn , which we assume here is locally constant. Let us fix a copy of R ⊂ Rn . Let us place a locally constant factorization algebra G on R, which we think of as the observables of a quantum mechanical system. We will view G as a homotopy-associative algebra or E1 algebra or A∞ algebra, but we will tend to write formulas as if everything were strictly associative. (One can always replace an A∞ algebra by a weakly equivalent but strictly associative dg algebra.) Let i∗ G denote the pushforward of G to a factorization algebra on Rn . Note that i∗ G takes value C on any open that does not intersect R, and it takes value G(i−1 (U)) on any open that does intersect R. Before we define a coupled factorization algebra, we need to understand the situation where the field theory on Rn and the field theory on R are uncoupled. In this case, the factorization algebra of observables for the system is simply F ⊗ i∗ G. We are interested in deforming this uncoupled situation. Definition 8.2.7 A coupling of our factorization algebra F to the factorization algebra G living on R is an element α ∈ F|R ⊗ G of cohomological degree 1 satisfying the Maurer–Cartan equation dα + 12 [α, α] = dα + α  α = 0, where  is the product in F|R ⊗ G viewed as an associative algebra. Here, by F|R , we mean the factorization algebra on R obtained by restricting F (e.g., by taking a small tubular neighborhood of this copy of R and then pushing forward to R). We can view F|R as a homotopy-associative algebra. The tensor product F|R ⊗ G is then also a homotopy-associative algebra. Remark: Bear in mind the following: (i) It is common to model homotopy associative algebras by A∞ algebras. If one takes this route, then one should use the L∞ version of the Maurer– Cartan equation.

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(ii) We are always imagining a situation in which our line defect is a small deformation of a decoupled system. One can capture this requirement by including a formal parameter, viewed as a coupling constant, next to α in this definition, but we will not do so. ♦ Given such a coupling α, we can deform the factorization algebra F|R ⊗ G by simply adding [α, −] to the differential of F|R ⊗ G. In other words, the twisted differential dα is given by dα (x) = dx + [α, x] = dx + α  x + x  α. We denote this factorization algebra on R by F|R ⊗α G. We can also deform F ⊗ i∗ G using this data. On R, it is simply F|R ⊗α G. Away from R, we change nothing and the factorization algebra is just given by F. It is not completely obvious that this construction makes sense, but we will see in the following lemma that it does. Lemma 8.2.8 This construction defines a factorization algebra F ⊗α i∗ G on Rn deforming F ⊗ i∗ G. We call this factorization algebra a line operator, and in the text that follows we will relate it explicitly to the standard example of a Wilson observable. Proof We will sketch a proof using technology developed in the literature on En algebras. For simplicity, let us suppose that R is a straight line in Rn . Let us identify Rn = R × Rn−1 and use coordinates (x, v) where x ∈ R and v ∈ Rn−1 . Our chosen copy of R is {v = 0}. Define a projection π : Rn \ R → R × R>0 (x, v) → (x, .v.). The pushforward π∗ F will be a locally constant factorization algebra on the two-dimensional manifold R × R>0 , and so an E2 algebra. We are interested in extensions of π∗ F into a factorization algebra on the half-space R × R≥0 , which is constructible with respect to the stratification where the two strata are R × 0 and R × R>0 . Such an extension will be an E1 algebra on R × 0, together with some compatibility between this E1 algebra and the E2 algebra π∗ F. We would like the E1 algebra on R×0 to be F|R ⊗α G. The work of Ayala, Francis, and Tanaka (2016) tell us that such an extension of π∗ F is the same as a constructible factorization algebra on Rn that is constructible with respect to the stratification where the strata are Rn \ R and R, and which on Rn \ R is F and on R is F|R ⊗α G. One can ask, what compatibility is required to glue the E2 algebra π∗ F on R × R>0 to the E1 algebra F|R ⊗α G? The answer is that we need to give

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a morphism of E2 algebras from π∗ F to the Hochschild cochain complex of F|R ⊗α G. Now, since the deformation of F|R ⊗ G given by [α, −] is an inner deformation, it does not change the Hochschild cochain complex (up to weak equivalence). Thus, the gluing data we need is the same as in the case when α = 0. But when α = 0, we already have the desired factorization algebra on Rn , namely F ⊗ i∗ G. Remark: It might seem restrictive to consider only deformations of F|R ⊗ G that are inner (i.e., given by an element α) rather than arbitrary deformations of the algebra structure. The last paragraph of the proof indicates one reason why we define “coupling” via inner deformations, and we provide some further motivations in the next section. ♦ Motivating this Construction There are two natural questions remaining from our discussion of line operators. First, in the case of Chern–Simons theory, how do we match up line operators with representations of the quantum group? And second, why does our notion of “coupling,” Definition 8.2.7, match up with the physical idea of coupling a quantum mechanical system to our field theory? Let us address the first point. In the factorization algebra approach, a onedimensional topological field theory is specified by an associative algebra, viewed as the algebra of operators (or observables) for the field theory. In other approaches to (topological) field theory Lurie (2009c), however, one is required to have a Hilbert space as well. A one-dimensional topological field theory is then specified by a finite-dimensional vector space V, which is the Hilbert space attached to a point. The algebra of operators is the matrix algebra End(V). It is thus natural to consider coupling a field theory to a topological quantum mechanical system of Atiyah–Segal–Lurie type, where the algebra of operators is End(V). If we do this, we find that a coupling is a Maurer–Cartan element in F|R ⊗ End(V). But such an element also provides a deformation of F|R ⊗ V as a free left F|R -module into a nontrivial projective left module. In this way, we have sketched why the following lemma is true. Lemma 8.2.9 The following are equivalent: (i) Specifying a projective rank k left module for the En algebra F. (ii) Coupling a locally constant factorization algebra F on Rn to the trivial quantum mechanical system on R ⊂ Rn whose algebra of operators is the matrix algebra glk .

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For Chern–Simons theory, this class of line operators matches up nicely with the usual Wilson observables. As a particularly simple example, let’s return to Abelian Chern–Simons theory. In that case, we know that F|R is equivalent to the algebra C[ε], where ε has cohomological degree one. Every degree one element in C[ε] ⊗ glk is of the form εT, with T ∈ glk , and every εT is a Maurer–Cartan element since the differential d here is trivial and ε2 = 0. A deformation of the action of C[ε] on C[ε] ⊗ V, of course, amounts to changing how ε acts, and we are free to change its action by left multiplication from the free action mε : M + εN → εM to any linear map of the form  mε : M + εN → εAM, where A ∈ glk . Now set T = A − 1. Next, let us explain how to interpret our concept of “coupling” from the point of view of physics. To do this, we need to say a little bit about local operators (that is, operators supported at a point) from the point of view of factorization algebras. This topic is treated in more detail in Volume 2. Here, we will be brief. In any field theory on Rn whose observables are given by a factorization algebra F, there is a co-stalk Fx of F for each point x ∈ Rn . This co-stalk is the limit limx∈U F(U) of the spaces of observables on neighborhoods U of x. The co-stalk Fx is a way to define local observables at x for the field theory: by definition an element of Fx is an observable contained in F(U) for every neighborhood U of x. For a general field theory, this limit might produce something hard to describe (and one might also worry about the difference between the limit and the homotopy limit). For a topological field theory, however, the limit is essentially constant, so that we can identify Fx with F(D) for any disc D around x. As x varies in Rn , the space Fx forms a vector bundle, typically of infinite rank. This vector bundle has a flat connection, reflecting the fact that we can differentiate local operators. Because Fx may be infinite-dimensional, one cannot in general solve the parallel transport equation from the flat connection. Thus, for a general factorization algebra, Fx should be thought of as a D-module. For a topological field theory, however, the bundle Fx is a local system, meaning one can solve the parallel transport equation. The equivalence Fx " Fy given by parallel transport arises by the quasi-isomorphisms "

"

Fx −→ F(D) ←− Fy for any disc D containing both x and y.

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Let us consider, as before, a line R ⊂ Rn , with a field theory on Rn whose observables are described by a factorization algebra F and with a field theory on R whose observables are described by a factorization algebra G. Let us assume that both factorization algebras are locally constant. For x ∈ R, we can consider the space Fx ⊗ Gx , which is the tensor product of local operators of our two field theories at x ∈ R ⊂ Rn . As we have seen, the space Fx ⊗ Gx forms, as x varies, a vector bundle with flat connection on R. Let us denote this vector bundle by Floc ⊗ Gloc . Since Floc ⊗ Gloc is a local system, there is a quasi-isomorphism F|R (D) ⊗ G(D) " ∗ (R, Floc ⊗ Gloc ) for any disc D ⊂ R. From a physics point of view, we might expect that to couple the system specified by F to that specified by G, we need to give a Lagrangian density on the real line R. In this case, such a Lagrangian would be a 1-form on R with values in the bundle of local observables Floc ⊗ Gloc , viewed as observables on the combined fields for the two field theories. Such a Lagrangian density does appear naturally in our earlier mathematical definition. Under the isomorphism F|R (D) ⊗ G(D) " ∗ (R, Floc ⊗ Gloc ), a Maurer–Cartan element α on the left-hand side corresponds to a sum Sα = Sα(0) + Sα(1) , (0)

where Sα is a function along R with values in the degree 1 component of the (1) vector bundle of local observables and Sα is a 1-form with values in the degree (1) 0 component of the local observables. The term Sα is thus a Lagrangian density. To understand better what it means, we need to go a bit further. Under the preceding correspondence, the Maurer–Cartan equation for α becomes the equation & % ddR Sα(0) + dF ⊗G Sα(0) + dF ⊗G Sα(1) + Sα(0) , Sα(1) = 0, where dF ⊗G is the differential on Floc ⊗ Gloc . This equation is essentially a version of the master equation in the Batalin–Vilkovisky (BV) formalism, but this relationship is manifest only with factorization algebras produced by BV quantization, which is the subject of Volume 2. Solving the equation perturbatively, however, we can get the flavor of the relationship.

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Consider the situation where we try to solve for Sα iteratively in some formal variable c:     (0) (1) (0) (1) + c2 Sα,2 + ··· . + Sα,1 + Sα,2 Sα = c Sα,1 At the first stage, we work modulo c2 . We then want to solve the equation (0)

(1)

ddR Sα,1 + dF ⊗G Sα,1 = 0. (0)

(Note that for grading reasons, we always need dF ⊗G Sα = 0.) And we are interested in solutions modulo the image of the operator ddR + dF ⊗G on elements in ∗ (R, Floc ⊗ Gloc ) of cohomological degree 0; this space is the firstorder version of gauge equivalence between solutions of the Maurer–Cartan equation. Unraveling these conditions, we see that we care about a Lagrangian density in Floc ⊗ Gloc up to a total derivative. These data are what one expects from physics. Continuing onto higher powers of c, one finds that our algebraic definition for coupling the quantum mechanical system G to the field theory F matches with what one expects from physics. In sum, for any topological field theory whose observables are given by a factorization algebra F on Rn , this discussion indicates that the category of projective left modules for F captures the category of Wilson line operators for the n-dimensional field theory described by F. These categories both admit interesting ways to “combine” or “fuse” objects: the left F-modules have an En−1 -monoidal structure, by Lurie’s work, and the line defects can be fused, equipping them with an En−1 -monoidal structure as well. (For a physically oriented introduction to the relationship between defects and higher categories, see Kapustin (2010).) We assert that these agree: when one examines the physical constructions, they behave as the homotopical algebra suggests. For some discussion of these issues, see Costello (2014). The Take-away Message Let us apply this discussion to Abelian Chern–Simons theory, where we have an E3 algebra ACS . The left modules that we find in our discussion above – projective modules of rank n – match exactly with the category LMod0 (ACS ) we introduced earlier. If we define the E2 -monoidal category of line operators of Abelian Chern–Simons theory to be the E2 -monoidal category LMod0 (ACS ), we find that our theorem relating Abelian Chern–Simons theory to the quantum group takes the following satisfying form. Theorem 8.2.10 The following are equivalent as braided monoidal categories:

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(i) The homotopy category of the category of Wilson line operators of Abelian Chern–Simons theory. (ii) The category of representations of the quantum group for the Abelian Lie algebra C. In this case, the identification between braided monoidal structures can be seen explicitly. We have already shown that the factorization algebra encodes the Wilson loop observables and recovers the Gauss linking number, but it is these relations that are used in providing the braiding on the line operators. (See, e.g., the discussion in Kapustin and Saulina (2011).)

8.2.4 Proof of a Technical Proposition Let us now give the proof of the following proposition, whose proof we skipped earlier. Proposition 8.2.11 View ACS as an E2 algebra via the map of operads from E2 to E3 . There is a canonical equivalence of E2 algebras from ACS to the commutative algebra C[ε], where the parameter ε is of degree 1. We view C[ε] as an E2 algebra via the map of operads from E2 to E∞ . Proof Recall that ACS is a twisted factorization envelope. Explicitly, it assigns to an open subset U ⊂ R3 , the cochain complex C∗ ( ∗c (U)[1] ⊕ C · h¯ [−1]) ⊗C[h¯ ] Ch¯ =1 . is given by the The Lie bracket on the dg Lie algebra ∗c (U)[1] ⊕ C · h[−1] ¯ shifted central extension of the Abelian Lie algebra ∗c (U)[1] for the cocycle  α ⊗ β → α ∧ β. We are interested in the restriction of this twisted enveloping factorization algebra to a plane R2 ⊂ R3 . Recall that ∗ iR2 ACS (U) = lim ACS (V), V⊂U

where the limit is taken over those opens V in R3 that contain our chosen open U ⊂ R2 . Note that since ACS is locally constant, this limit is eventually constant. If we identify R3 = R2 × R, we can identify ∗ iR2 ACS (U) = ACS (U × R). In other words, we can identify i∗R2 ACS with the pushforward π∗ ACS where π : R3 → R2 is the projection. This identification is as factorization algebras.

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There is a quasi-isomorphism, for all U ⊂ R2 , ∗c (U)[−1] " ∗c (U × R). Concretely, we can define this cochain map as follows. Choose a function f on R with compact supprt such that f (t) dt = 1. The quasi-isomorphism sends a form ω ∈ ∗c (U) to ω ∧ f (t) dt, where t indicates the coordinate on R. The factorization algebra π∗ ACS is the twisted factorization algebra of the precosheaf of Abelian dg Lie algebras π∗ ∗c [1]. To an open subset U ⊂ R2 , this Abelian Lie algebra assigns ∗c (U×R)[1]. The cocycle defining the central extension is given by the formula  [α, β] = h¯

α ∧ β. U×R

We can restrict this cocycle to one on the sub-cosheaf ∗c (U), where we use the map ∗c (U) → ∗c (U × R)[1] discussed earlier. We find that it is zero, since for ω1 , ω2 ∈ ∗c (U)[1], we have  ω1 f (t)dt ∧ ω2 f (t) dt = 0. U×R

It follows that we have a quasi-isomorphism of precosheaves of Lie algebras on R2 : C · h[−1] ⊕ ∗c (U) " C · h¯ [−1] ⊕ ∗c (U × R)[1]. ¯ On the right hand side, the central extension is the one defined above, and on the left-hand side it is zero. By applying the twisted factorization envelope construction to both sides, we find that that π∗ ACS is quasi-isomorphic to the un-twisted factorization envelope of the precosheaf of Abelian dg Lie algebras ∗c on R2 . Since this Lie algebra is Abelian, and there is no central extension, the result is a commutative factorization algebra. The corresponding E2 algebra must therefore come from a commutative algebra. Let us now identify this E2 algebra explicitly. Note that, for every open U ⊂ R2 , there is a cochain map ∗c (U) → C[−2]

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given by integration over U. This map is a quasi-isomorphism if U is a disc. It also respects the cosheaf operations, and so gives a map of cosheaves on R2 ∗c → C[−2] to the constant cosheaf C[−2]. The constant cosheaf C assigns to any open U ⊂ R2 the space H0 (U, C), which is a coproduct of a copy of C for each connected component of U. Applying the factorization envelope construction, we find a map of prefactorization algebras C∗ ( ∗c ) = Sym ∗c [1] → Sym C[−1]. This map is a quasi-isomorphism when restricted to disjoint unions of discs. It therefore identifies the E2 algebra with the commutative algebra Sym C[−1] = C[ε]. Remark: There is another approach to this lemma that is less direct but applicable more broadly, by using the machinery of deformation theory. There is a Hochschild complex for En algebras (in cochain complexes over a field of characteristic zero) akin to the Hochschild cochain complex for associative algebras. Furthermore, given an En algebra A, there is an exact sequence of the form Def(A) → HHE∗n (A) → A relating the En Hochschild complex of A and the complex describing En algebra deformations of A (see Francis (2013)). First-order deformations of the En algebra structure are given by the n + 1st cohomology group of the complex Def(A). The funny shift is due to the fact that the En deformation complex is an En+1 algebra and the underlying dg Lie algebra is obtained by shifting to turn the Poisson bracket into an unshifted Lie bracket. Compare, for instance, the fact that HH 2 (A) classifies first-order deformations of associative algebras. Moreover, there is an analog of the Hochschild–Kostant–Rosenberg theorem for En algebras: for A a commutative dg algebra A, it identifies the En deformation complex for A with the shifted polyvector fields SymA (TA [−n]). Here TA denotes the tangent complex of A. (See Calaque and Willwacher (2015) for the full statement and a proof.) For us, the classical observables of Abelian Chern–Simons are equivalent to the commutative dg algebra C[ε], where ε has cohomological degree 1, because it is the factorization envelope of an Abelian Lie algebra. The E2 HKR theorem then tells us that ) * HHE∗2 (C[ε]) " C ε, ∂/∂ε ,

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where ε and ∂/∂ε have cohomological degree 1. There are no degree 3 elements of our deformation complex, so the E2 deformation must be trivial. Hence ACS is a trivial deformation as an E2 algebra. Note, by contrast, that there is a one-dimensional space of deformations of C[ε] as an E3 ) first-order * algebra because HHE∗3 (C[ε]) is C ε, ∂/∂ε , with ∂/∂ε now in degree 2. ♦

Appendix A Background

We use techniques from disparate areas of mathematics throughout this book and not all of these techniques are standard knowledge, so here we provide a terse introduction to • Simplicial sets and simplicial techniques • Operads, colored operads (or multicategories), and algebras over colored operads • Differential graded (dg) Lie algebras and their (co)homology • Sheaves, cosheaves, and their homotopical generalizations • Elliptic complexes, formal Hodge theory, and parametrices along with pointers to more thorough treatments. By no means do readers need to be expert in all these areas to use our results or follow our arguments. They just need a working knowledge of this background machinery, and this appendix aims to provide the basic definitions, to state the results relevant for us, and to explain the essential intuition. We do assume that readers are familiar with basic homological algebra and basic category theory. For homological algebra, there are numerous excellent sources, in books and online, among which we recommend the complementary texts by Weibel (1994) and Gelfand and Manin (2003). For category theory, the standard reference Mac Lane (1998) is adequate for our needs; we also recommend the series Borceux (1994a). Remark: Our references are not meant to be complete, and we apologize in advance for the omission of many important works. We simply point out sources that we found pedagogically oriented or particularly accessible. ♦

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Background

A.1 Reminders and Notation We overview some terminology and notations before embarking on our expositions. For C a category, we often use x ∈ C to indicate that x is an object of C. We typically write C(x, y) to the denote the set of morphisms between the objects x and y, although occasionally we use HomC (x, y). The opposite category C op has the same objects but C op (x, y) = C(y, x). Given a collection of morphisms S in C, a localization of C with respect to S is a category C[S−1 ] and a functor q : C → C[S−1 ] satisfying the following conditions. (i) For every morphism s ∈ S, its image q(s) is an isomorphism. (ii) For any functor F : C → D such that F(s) is an isomorphism for every s ∈ S, there is a functor F : C[S−1 ] → C and a natural isomorphism τ : F ◦ F ⇒ F. (iii) For every category D, the functor q∗ : Fun(C[S−1 ], D) → Fun(C, D) sending G to G ◦ q is full and faithful. These ensure that the localization is unique up to equivalence of categories, if it exists. (We do not discuss size issues in this book.) The localization is the category in which every s ∈ S becomes invertible. A morphism f : C[S−1 ](x, y) can be concretely understood as a “zig zag” "

"

x ← z1 → z2 ← · · · → y where the arrows going to the left (the “wrong way”) are all in S and the arrows going to the right are arbitrary morphisms from C. We typically apply localization in the setting of a category C with a class W of weak equivalences. We call W a class of weak equivalences if W contains all isomorphisms in C and satisfies the two-out-of-three property, which states that given morphisms f ∈ C(x, y) and g ∈ C(y, z) such that two morphisms in the set {f , g, g ◦ f } are in W, then all three are in W. As examples, consider (i) Any category with isomorphisms as weak equivalences. (ii) The category of cochain complexes with cochain homotopy equivalences as weak equivalences. (iii) The category of cochain complexes with quasi-isomorphisms as weak equivalences. (iv) The category of topological spaces with weak homotopy equivalences as weak equivalences. We often denote the localization C[W −1 ] by Ho(C, W), or just Ho(C), and call it the “homotopy category.”

A.2 Simplicial Techniques

275

A.2 Simplicial Techniques Simplicial sets are a combinatorial substitute for topological spaces, so it should be no surprise that they can be quite useful. On the one hand, we can borrow intuition for them from algebraic topology; on the other, simplicial sets are extremely concrete to work with because of their combinatorial nature. In fact, many constructions in homological algebra are best understood via their simplicial origins. Analogs of simplicial sets (i.e., simplicial objects in other categories) are useful as well. In this book, we use simplicial sets in two ways: • When we want to talk about a family of quantum field theories (QFTs) or space of parametrices, we will usually construct a simplicial set of such objects instead. • We accomplish some homological constructions by passing through simplicial sets (e.g., in constructing the extension of a factorization algebra from a basis). After giving the essential definitions, we state the main theorems we use. Definition A.2.1 Let denote the category whose objects are totally ordered finite sets and whose morphisms are nondecreasing maps between ordered sets. We usually work with the skeletal subcategory whose objects are [n] = {0 < 1 < · · · < n}. A morphism f : [m] → [n] then satisfies f (i) ≤ f (j) if i < j. We will relate these objects to geometry in the text that follows, but it helps to bear in mind the following picture. The set [n] corresponds to the n-simplex n equipped with an ordering of its vertices, as follows. View the n-simplex n as living in Rn+1 as the set ⎧ ⎫ +  ⎨ ⎬ + (x0 , x1 , . . . , xn ) ++ xj = 1 and 0 ≤ xj ≤ 1 ∀j . ⎩ ⎭ j

Identify the element 0 ∈ [n] with the 0th basis vector e0 = (1, 0, . . . , 0), 1 ∈ [n] with e1 = (0, 1, 0, . . . , 0), and k ∈ [n] with the kth basis vector ek . The ordering on [n] then prescribes a path along the edges of n , starting at e0 , then going to e1 , and continuing till the path ends at en . Every map f : [m] → [n] induces a linear map f∗ : Rm+1 → Rn+1 by setting f∗ (ek ) = ef (k) . This linear map induces a map of simplices f∗ : m → n . There are particularly simple maps that play an important role throughout the subject. Note that every map f factors into a surjection followed by an

276

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injection. We can then factor every injection into a sequence of coface maps, namely maps of the form fk : [n] i

→



→

[n + 1] i, i≤k i + 1, i > k

.

Similarly, we can factor every surjection into a sequence of codegeneracy maps, namely maps of the form dk : [n] i

→ →



[n − 1] i, i≤k i − 1, i > k

.

The names face and degeneracy fit nicely with the picture of the geometric simplices: a coface map corresponds to a choice of n-simplex in the boundary of the n + 1-simplex, and a codegeneracy map corresponds to “collapsing” an edge of the n-simplex to project the n-simplex onto an n − 1-simplex. We now introduce the main character. Definition A.2.2 A simplicial set is a functor X : op → Set, often denoted X• . The set X([n]), often denoted Xn , is called the “set of n-simplices of X.” A map of simplicial sets F : X → Y is a natural transformation of functors. We denote this category of simplicial sets by sSet. Let’s quickly examine what the coface maps tell us about a simplicial set X. For example, by definition, a map fk : [n] → [n + 1] in goes to X(fk ) : Xn+1 → Xn . We interpret this map X(fk ) as describing the kth n-simplex sitting as a “face on the boundary” of an n + 1-simplex of X. A similar interpretation applies to the dk .

A.2.1 Simplicial Sets and Topological Spaces When working in a homotopical setting, simplicial sets often provide a more tractable approach than topological spaces themselves. In this book, for instance, we describe “spaces of field theories” as simplicial sets. In the text that follows, we sketch how to relate these two kinds of objects. One can use a simplicial set X• as the “construction data” for a topological space: each element of Xn labels a distinct n-simplex n , and the structure maps of X• indicate how to glue the simplices together. In detail, the geometric realization is the quotient topological space   . n Xn ×  / ∼ |X• | = n

A.2 Simplicial Techniques

277

under the equivalence relation ∼ where (x, s) ∈ Xm × m is equivalent to (y, t) ∈ Xn × n if there is a map f : [m] → [n] such that X(f ) : Xn → Xm sends y to x and f∗ (s) = t. Lemma A.2.3 Under the Yoneda embedding, [n] defines a simplicial set

[n] : [m] ∈ op → ([m], [n]) ∈ Sets. The geometric realization of [n] is the n-simplex n (more accurately, it is homeomorphic to the n-simplex). In general, every (geometric) simplicial complex can be obtained by the geometric realization of some simplicial set. Thus, simplicial sets provide an efficient way to study combinatorial topology. One can go the other way, from topological spaces to simplicial sets: given a topological space X, there is a simplicial set Sing X, known as the “singular simplicial set of S.” The set of n-simplices (Sing X)n is simply Top(n , X), the set of continuous maps from the n-simplex n into X. The structure maps arise from pulling back along the natural maps of simplices. Theorem A.2.4 Geometric realization and the singular functor form an adjunction | − | : sSet  Top : Sing between the category of simplicial sets and the category of topological spaces. Thus, when we construct a simplicial set of Batalin–Vilkovisky (BV) theories T• , we obtain a topological space |T• |. This theorem suggests as well how to transport notions of homotopy to simplicial sets: the homotopy groups of a simplicial set X• are the homotopy groups of its realization |X• |, and maps of simplicial sets are homotopic if their realizations are. We would then like to say that these functors | − | and Sing make the categories of simplicial sets and topological spaces equivalent, in some sense, particularly when it comes to questions about homotopy type. One sees immediately that some care must be taken, since there are topological spaces very different in nature from simplicial or cell complexes and for which no simplicial set could provide an accurate description. The key is only to think about topological spaces and simplicial sets up to the appropriate notion of equivalence. Remark: It is more satisfactory to define these homotopy notions directly in terms of simplicial sets and then to verify that these match up with the topological notions. We direct readers to the references for this story, as the details are not relevant to our work in the book. ♦

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Background

We say a continuous map f : X → Y of topological spaces is a weak equivalence if it induces a bijection between connected components and an isomorphism πn (f , x) : πn (X, x) → πn (Y, f (x)) for every n > 0 and every x ∈ X. Let Ho(Top), the homotopy category of Top, denote the category of topological spaces where we localize at the weak equivalences. There is a concrete way to think about this homotopy category. For every topological space, there is some CW complex weakly equivalent to it, under a zig zag of weak equivalences; and by the Whitehead theorem, a weak equivalence between CW complexes is in fact a homotopy equivalence. Thus, Ho(Top) is equivalent to the category of CW complexes with morphisms given by continuous maps modulo homotopy equivalence. Likewise, let Ho(sSet) denote the homotopy category of simplicial sets, where we localize at the appropriate notion of simplicial homotopy. Then Quillen proved the following wonderful theorem. Theorem A.2.5 The adjunction induces an equivalence | − | : Ho(sSet)  Ho(Top) : Sing between the homotopy categories. (In particular, they provide a Quillen equivalence between the standard model category structures on these categories.) This theorem justifies the assertion that, from the perspective of homotopy type, we are free to work with simplicial sets in place of topological spaces. In addition, it helpful to know that algebraic topologists typically work with a better behaved categories of topological spaces, such as compactly generated spaces. Among simplicial sets, those that behave like topological spaces are known as Kan complexes or fibrant simplicial sets. Their defining property is a simplicial analogue of the homotopy lifting property, which we now describe explicitly. The horn for the kth face of the n-simplex, denoted k [n], is the subsimplicial set of [n] given by the union of the all the faces [n − 1] → [n] except the kth. (As a functor on , the horn takes the [m] to monotonic maps [m] → [n] that do not have k in the image.) A simplicial set X is a Kan complex if for every map of a horn k [n] into X, we can extend the map to the n-simplex [n]. Diagrammatically, we can fill in the dotted arrow k [n]  z

[n]

z

z

/X z=

A.2 Simplicial Techniques

279

to get a commuting diagram. In general, one can always find a “fibrant replacement” of a simplicial set X• (e.g., by taking Sing |X• | or via Kan’s Ex∞ functor) that is weakly equivalent and a Kan complex.

A.2.2 Simplicial Sets and Homological Algebra Our other use for simplicial sets relates to homological algebra. We always work with cochain complexes, so our conventions will differ from those who prefer chain complexes. For instance, the chain complex computing the homology of a topological space is concentrated in nonnegative degrees. We work instead with the cochain complex concentrated in nonpositive degrees. (To convert, simply swap the sign on the indices.) Definition A.2.6 A simplicial Abelian group is a simplicial object A• in the category of Abelian groups, i.e., a functor A : op → Ab. By composing with the forgetful functor U : Ab → Set that sends a group to its underlying set, we see that a simplicial Abelian group has an underlying simplicial set. To define simplicial vector spaces or simplicial R-modules, one simply replaces Abelian group by vector space or R-module everywhere. All the work below will carry over to these settings in a natural way. There are two natural ways to obtain a cochain complex of Abelian groups (respectively, vector spaces) from a simplicial abelian group. The unnormalized chains CA• is the cochain complex  A|m| , m ≤ 0 (CA• )m = 0, m>0 with differential d : (CA• )m a

→ →

(CA• )m+1 k k=0 (−1) A• (fk )(a),

|m|

where the fk run over the coface maps from [|m| − 1] to [|m|]. The normalized chains NA• is the cochain complex (NA• )m =

|m|−1
1. ♦ Similarly, a K-linear category is a multicategory over the symmetric monoidal category (VectK , ⊗K ) with no compositions between two or more elements. Example: An operad O, in the sense of the preceding subsection, is a multicategory over the symmetric monoidal category (VectK , ⊗K ) with a single object ∗ and O(∗, . . . , ∗ | ∗) = O(n).    n

Replacing VectK with another symmetric monoidal category C, we obtain a definition for operad in C. ♦

A.4 Lie Algebras and Chevalley–Eilenberg Complexes

291

Example: Every symmetric monoidal category (C, ) has an underlying multicategory C with the same objects and with maps C(x1 , . . . , xn | y) = C(x1  · · ·  xn , y). When C = VectK , these are precisely all the multilinear maps.

♦

For every multicategory M, one can construct a symmetric monoidal envelope SM by forming the left adjoint to the “forgetful” functor from symmetric monoidal categories to multicategories. An object of SM is a formal finite sequence [x1 , . . . , xm ] of colors xi , and a morphism f : [x1 , . . . , xm ] → [y1 , . . . , yn ] consists of a surjection φ : {1, . . . , m} → {1, . . . , n} and a morphism fj ∈ M({xi }i∈φ −1 (j) | yj ) for every 1 ≤ j ≤ n. The symmetric monoidal product in SM is simply concatenation of formal sequences. Finally, an algebra over a colored operad M with values in N is simply a functor of multicategories F : M → N . When we view O as a multicategory and use the underlying multicategory VectK , then F : O → VectK reduces to an algebra over the operad O as in the preceding subsection.

References For a readable discussion of operads, multicategories, and different approaches to them, see Leinster (2004). Beilinson and Drinfeld (2004) develop pseudotensor categories – yet another name for this concept – for exactly the same reasons as we do in this book. Lurie (2016) provides a thorough treatment of colored operads compatible with ∞-categories.

A.4 Lie Algebras and Chevalley–Eilenberg Complexes Lie algebras, and their homotopical generalization L∞ algebras, appear throughout both volumes in a variety of contexts. It might surprise readers that we never use their representation theory or almost any aspects emphasized in textbooks on Lie theory. Instead, we primarily use dg Lie algebras as a convenient language for describing deformations of some structure, which is a central theme of Volume 2. For the purposes of Volume 1, we need to describe standard homological constructions with dg Lie algebras. For the definition of L∞ algebras and the relationship with derived geometry, see Volume 2.

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Background

A.4.1 A Quick Review of Homological Algebra Lie Algebras Let K be a field of characteristic zero. (We always have in mind K = R or C.) A Lie algebra over K is a vector space g with a bilinear map [−, −] : g⊗K g → g such that • [x, y] = −[y, x] (antisymmetry) and • [x, [y, z]] = [[x, y], z] + [y, [x, z]] (Jacobi rule). The Jacobi rule is the statement that [x, −] acts as a derivation. A simple example is the space of n × n matrices Mn (K), usually written as gln in this context, where the bracket is [A, B] = AB − BA, the commutator using matrix multiplication. Another classic example is given by Vect(M), the vector fields on a smooth manifold M, via the commutator bracket of derivations acting on smooth functions. A module over g, or representation of g, is a vector space M with a bilinear map ρ : g ⊗K M → M such that ρ(x ⊗ ρ(y ⊗ m)) − ρ(y ⊗ ρ(x ⊗ m)) = ρ([x, y] ⊗ m). Usually, we will suppress the notation ρ and simply write x · m or [x, m]. Continuing with the examples from above, the matrices gln acts on Kn by left multiplication, so Kn is naturally a gln -module. Analogously, vector fields Vect(M) act on smooth functions C∞ (M) as derivations, and so C∞ (M) is a Vect(M)module. There is a category g − mod whose objects are g-modules and whose morphisms are the natural structure-preserving maps. To be explicit, a map f ∈ g − mod(M, N) of g-modules is a linear map f : M → N such that [x, f (m)] = f ([x, m]) for every x ∈ g and every m ∈ M. Lie algebra homology and cohomology arise as the derived functors of two natural functors on the category of g-modules. We define the invariants as the functor (−)g : g − mod M

→ →

VectK Mg

where M g = {m | [x, m] = 0 ∀x ∈ g}. (A nonlinear analog is taking the fixed points of a group action on a set.) The coinvariants is the functor (−)g : g − mod M

→ →

VectK Mg

A.4 Lie Algebras and Chevalley–Eilenberg Complexes

293

where Mg = M/gM = M/{[x, m] | x ∈ g, m ∈ M}. (A nonlinear analog is taking the quotient, or orbit space, of a group action on a set, i.e., the collection of orbits.) To define the derived functors, we rework our constructions into the setting of modules over associative algebras so that we can borrow the Tor and Ext functors. The universal enveloping algebra of a Lie algebra g is Ug = Tens(g)/(x ⊗ y − y ⊗ x − [x, y]) where Tens(g) = ⊕n≥0 g⊗n denotes the tensor algebra of g. Note that the ideal by which we quotient ensures that the commutator in Ug agrees with the bracket in g: for all x, y ∈ g, x · y − y · x = [x, y], where · denotes multiplication in Ug. It is straightforward to verify that there is an adjunction U : LieAlgK  AssAlgK : Forget, where Forget(A) views an associative algebra A over K as a vector space with bracket given by the commutator of its product. As a consequence, we can view g − mod as the category of left Ug-modules, which we’ll denote Ug − mod, without harm. Now observe that K is a trivial g-module for any Lie algebra g: x · k = 0 for all x ∈ g and k ∈ K. Moreover, K is the quotient of Ug by the ideal (g) generated by g itself, so that K is a bimodule over Ug. It is then straightforward to verify that Mg = K ⊗Ug M

and

M g = HomUg (K, M)

for every module M. Definition A.4.1 For M a g-module, the Lie algebra homology of M is Ug

H∗ (g, M) = Tor∗ (K, M), and the Lie algebra cohomology of M is H ∗ (g, M) = Ext∗Ug (K, M). Notice that H 0 and H0 recover invariants and coinvariants, respectively. There are concrete interpretations of the lower (co)homology groups, such as that H 1 (g, M) describes “outer derivations” (i.e., derivations modulo inner derivations) and and H 2 (g, M) describes extensions of g as a Lie algebra by M. There are standard cochain complexes for computing Lie algebra (co)homology, and their generalizations will play a large role throughout the

294

Background

book. The key is to find an efficient, tractable resolution of K as a Ug module. We use again that K is a quotient of Ug to produce a resolution: "  · · · → ∧n g ⊗K Ug → · · · → g ⊗K Ug → Ug −→ K, where the final map is given by the quotient and the remaining maps are of the form (y1 ∧ · · · ∧ yn ) ⊗ (x1 · · · xm ) →

n  (−1)n−k (y1 ∧ · · · yk · · · ∧ yn ) k=1

⊗ (yk · x1 · · · xm )  (−1)j+k−1 ([yj , yk ] ∧ y1 ∧ · · ·  yj · · · yk · · · ∧ yn ) − 1≤j m. Let Seq denote the functor category Fun(Z≤0 , A). We call an object · · · X(n) → X(n + 1) → · · · → X(0)

C.5 Homotopy Colimits

371

in Seq a sequence. Equip Seq with the projective model structure: a map of sequences f : X → Y is • A weak equivalence if f (n) : X(n) → Y(n) is a quasi-isomorphism for every n. • A fibration if f (n) : X(n) → Y(n) is a fibration for every n. Note that we are working with levelwise weak equivalences, which we denote WL . The following observation gives a nice explanation for the role of filtered cochain complexes: they are the cofibrant sequences. Lemma C.5.12 A sequence X is cofibrant if and only if every map X(n) → X(n+1) is a cofibration (i.e., a monomorphism in every cohomological degree). Hence, we introduce the following definition. Definition C.5.13 Let Afil denote the category Seqc of cofibrant objects, i.e., the filtered cochain complexes. We want to work with a different notion of weak equivalence than WL . We say a map of sequences f : X → Y is a filtered weak equivalence if the induced maps hocofib(X(n) → X(n + 1)) → hocofib(Y(n) → Y(n + 1)) are quasi-isomorphisms for every n. Here “hocofib” is the homotopical version of the cokernel, and it can be computed using the standard cone construction of homological algebra. This condition is the homotopical version of saying we have a quasi-isomorphism on the associated graded, since we are taking the homotopy quotient rather than the naive quotient. On the filtered cochain complexes, it agrees with our earlier notion of a filtered weak equivalence, since the homotopy cofiber agrees with the cokernel. (This observation gives an explanation for why one should work with filtered complexes: one can do the naive computations.) Let WF denote the collection of filtered weak equivalences. We want to get a handle on the ∞-category Afil [WF−1 ], which can be called the filtered derived ∞-category of A. The complete filtered cochain complexes will be the crucial tool for us. Recall that a filtered cochain complex X ∈ Afil is complete if the canonical map X(0) → limn X(0)/X(n) is a quasi-isomorphism. (We apologize for the slight change of notation from Fn X to X(n), but this notation makes clearer the comparison with sequences, which is useful in this context.)

372

Homological Algebra in Differentiable Vector Spaces

Definition C.5.14 Let Acfil denote the category of completed filtered cochain complexes in A (i.e., the pro-cochain complexes in A, in analogy with the terminology of Section C.4). The inclusion functor Acfil → Afil is right adjoint to a “completion functor” that sends a filtration X = (· · · → X(n) → X(n + 1) → · · · → X(0)) to its completion  X where  X (0) = limn X(0)/X(n) equipped with the filtration  X (n) = ker( X → X(0)/X(n)). Note that completion is “idempotent” in the sense that a complete filtered cochain complex is isomorphic to its completion. Lemma C.5.15 The inclusion (Acfil , WF ) → (Afil , WF ) induces an equivalence between their Dwyer–Kan simplicial localizations, and hence we have an equivalence Acfil [WF−1 ] " Afil [WF−1 ] between the ∞-categories presented by these categories with weak equivalences. Proof Under this inclusion-completion adjunction, we see that filtered weak equivalences are preserved. Moreover, for every filtered cochain complex X, the canonical map X →  X is a filtered weak equivalence. By Corollary 3.6 of Dwyer and Kan (1980a), these have weakly equivalent hammock localizations, which implies that the standard simplicial localizations are weakly equivalent.

The Theorem  I (F) of a diagram Definition C.5.16 The completed cone construction Cone  ⊕ F : I → Acfil is given by Tot (CI (F)), the totalization (via the completed direct sum) of the double complex CI (F) of Section C.5.2. Theorem C.5.17 Let F : I → Acfil be a diagram. The completed cone con I (F) is a model for the colimit of F in the ∞-category Acfil [W −1 ]. struction Cone F Proof We can view this diagram as living in Afil (and hence also the category Seq), and by Lemma C.5.15, it suffices to compute the colimit in Afil [WF−1 ]. We will thus verify the (uncompleted) cone construction provides a model for this colimit. Completing this cone will provide a complete filtered cochain complex that is filtered weak equivalent, and hence also a model for the desired colimit. We compute this colimit in filtered cochain complexes by using model categories. Let SeqL denote Seq equipped with the projective model structure, so

C.5 Homotopy Colimits

373

the weak equivalences are WL . Observe that WL ⊂ WF as collections of morphisms inside Seq. Then the left Bousfield localization at WF , which we will denote SeqF , has the same cofibrations, by definition. Hence, if one wants to compute homotopy colimits in SeqF , they agree with the homotopy colimits in SeqL . (More explicitly, the projective model structures on Fun(I, SeqF ) and Fun(I, SeqL ) have the same cofibrant replacement functor, so homotopy colimits can be computed in either category.) In SeqL , our computation becomes simple: a homotopy colimit of diagram in such a functor category can be computed objectwise, so we can apply Theorem C.5.11 objectwise. This construction naturally produces ConeI with the natural filtration.

Appendix D The Atiyah–Bott Lemma

Atiyah and Bott (1967) shows that for an elliptic complex (E , Q) on a compact closed manifold M, with E the smooth sections of a Z-graded vector bundle, there is a homotopy equivalence (E , Q) → (E , Q) into the elliptic complex of distributional sections. The argument follows from the existence of parametrices for elliptic operators. This result was generalized to the noncompact case in Tarkhanov (1987). Let M be a smooth manifold, not necessarily compact. Let (E , Q) be an elliptic complex on M. Let E denote the complex of distributional sections of E . We will endow both E and E with their natural topologies. Lemma D.1 (Lemma 1.7, Tarkhanov (1987)) The natural inclusion (E , Q) → (E , Q) admits a continuous homotopy inverse. The continuous homotopy inverse :E →E is given by a kernel K ∈ E ! ⊗ E with proper support. The homotopy S : E → E is a continuous linear map with [Q, S] =  − Id . !

The kernel KS for S is a distribution, that is, an element of E ⊗ E with proper support. Proof We will reproduce the proof in Tarkhanov (1987). Choose a metric on E and a volume form on M. (If E is a complex vector bundle, use a Hermitian metric.) Let Q∗ be the formal adjoint to Q, which is a degree −1 differential operator, and form the graded commutator D = [Q, Q∗ ]. This operator D is elliptic on each space E i , the cohomological degree i part of E . Thus, by standard results in the theory of pseudodifferential operators, there is a parametrix 374

The Atiyah–Bott Lemma

375

!

P for D. The kernel KP is an element of E ⊗E , and the corresponding operator P : Ec → E is an inverse for D up to smoothing operators. By multiplying KP by a smooth function on M × M that is 1 in a neighborhood of the diagonal, we can assume that KP has proper support. Hence, we can extend P to a map P : E → E that is still a parametrix: thus P ◦ D and D ◦ P both differ from the identity by smoothing operators. The homotopy S is now defined by S = Q∗ P. Note that the homotopy inverse  : E → E and the homotopy S : E → E produce a cochain homotopy equivalence in DVS, after applying the functor dift : LCTVS → DVS. The following corollary is quite useful in the study of free field theories. Let Ec denote the cosheaf that assign to an open U ⊂ M the cochain complex of differentiable vector spaces (Ec (U), Q), namely the compactly supported smooth sections on U of the graded vector bundle E. Note that, somewhat abusively, we are viewing these as differentiable vector spaces rather than topological vector spaces. Similarly, let E c denote the cosheaf sending U to (E c (U), Q), namely the compactly supported distributional sections on U of the graded vector bundle E, viewed as a differentiable cochain complex. (See Section B.7.2 in Appendix B and Lemma 6.5.2 for the proof that we have a cosheaf.) The canonical map Ec → E c is a quasi-isomorphism of cosheaves of differentiable cochain complexes. Proof The Atiyah–Bott–Tarkhanov lemma assures us that Ec (U) → E c (U) is a quasi-isomorphism on every open U.

References

Ad´amek, Jiˇr´ı, and Rosick´y, Jiˇr´ı. 1994. Locally presentable and accessible categories. London Mathematical Society Lecture Note Series, Vol. 189. Cambridge: Cambridge University Press. Atiyah, M., and Bott, R. 1967. A Lefschetz fixed point formula for elliptic complexes: I. Ann. Math., (2), 86(2), 374–407. ´ Atiyah, Michael. 1988. Topological quantum field theories. Inst. Hautes Etudes Sci. Publ. Math., 175–186 (1989). Ayala, David, and Francis, John. 2015. Factorization homology of topological manifolds. J. Topol., 8(4), 1045–1084. Ayala, David, Francis, John, and Rozenblyum, Nick. 2015. Factorization homology from higher categories. Available at http://arxiv.org/abs/1504.04007. Ayala, David, Francis, John, and Tanaka, Hiro Lee. 2016. Factorization homology for stratified manifolds. Available at http://arxiv.org/abs/1409.0848. Behnke, Heinrich, and Stein, Karl. 1949. Entwicklung analytischer Funktionen auf Riemannschen Fl¨achen. Math. Ann., 120, 430–461. Beilinson, Alexander, and Drinfeld, Vladimir. 2004. Chiral algebras. American Mathematical Society Colloquium Publications, Vol. 51. Providence, RI: American Mathematical Society. Ben Zvi, David, Brochier, Adrien, and Jordan, David. 2016. Integrating quantum groups over surfaces: Quantum character varieties and topological field theory. Available at http://arxiv.org/abs/1501.04652. Boardman, J. M., and Vogt, R. M. 1973. Homotopy invariant algebraic structures on topological spaces. Lecture Notes in Mathematics, Vol. 347. Berlin and New York: Springer-Verlag. Boavida de Brito, Pedro, and Weiss, Michael. 2013. Manifold calculus and homotopy sheaves. Homol. Homot. Appl., 15(2), 361–383. B¨odigheimer, C.-F. 1987. Stable splittings of mapping spaces. Algebraic topology (Seattle, Wash., 1985), pp. 174–187. Lecture Notes in Mathematics, Vol. 1286. Berlin: Springer-Verlag. Borceux, Francis. 1994a. Handbook of categorical algebra. 1. Encyclopedia of Mathematics and Its Applications, Vol. 50. Cambridge: Cambridge University Press. Basic category theory.

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Index

BD algebra, 106, 287 BD operad, 287 C∗ (g), 295 C∗ (g), 294 En algebra, 14, 135, 221, 250 as locally constant factorization algebra, 221 En -monoidal category, 254 P0 algebra, 106, 287 P0 operad, 287 R-matrix, 256 BVS, 310 CVS, 310 DVS, 310 self-enrichment of, 337

, 275 E , 68, 314 Ec , 68, 314 LCTVS, 310 S Disj, 49 βγ system, 171 factorization homology of, 248 E , 68, 316 E c , 68, 316 D, 316 C∞ -module, 62, 312 distributional sections of a vector bundle, 316 distributions, 315 holomorphic sections of a holomorphic vector bundle, 317 smooth sections of a vector bundle, 314 from a topological vector space, 326 Disj, 47

DisjG , see also prefactorization algebra, equivariant Mfld, 62, 311 PDiscs, 150

G , see also prefactorization algebra, Disj equivariant ˇ Cech complex, 212, 299 Discsn , 135 action functional, 24, 41 as differential on observables, 97 algebra of functions on sections of a vector bundle, 349 associative algebra in quantum mechanics, 6 Atiyah–Bott lemma, 95, 374 Batalin–Vilkovisky formalism, 23, 106, 287 bornological vector space, 329 bornology, 328 braided monoidal category, 249 BV formalism, see Batalin–Vilkovisky formalism canonical quantization, see quantization, canonical central extension, see local Lie algebra Chern–Simons theory, see field theory, Abelian Chern-Simons chiral algebra, 11 chiral conformal field theory, see field theory, chiral conformal chiral homology, 11

383

384

Index

cluster decomposition principle, 140 cobordism, 12, 14, 220 Cobordism Hypothesis, 220 colimit homotopy, see homotopy colimit compactification, 112 cone, 363 completed, 372 convenient vector space, 67, 332 sections of vector bundles as, 334 tensor product of, 335 cooperad, 287 correlation function, 14, 129, 140 correlator, 37 correspondence principle, 6 cosheaf, 298 gluing axiom, 208 homotopy, 211, 299 cosimplicial cobar construction, 367 coupling, 263 cover Weiss, 209

defect, 249, 262 point, 52 dense inclusion, 67 derived critical locus, 91 descendant, 154 descent along a torsor, 241 differentiable cochain complex, 64 complete filtered, 355 filtered, 355 differentiable pro-cochain complex, 357 completed direct sum, 358 differentiable vector space, 63, 313 distributions, 316 map of, 314 self-enrichment, 337 smooth sections of a vector bundle, 314 stalk of, 317 differential complex, 305 divergence complex, 90 divergence operator, 25, 29, 89

elliptic complex, 306 elliptic operator, 305 enveloping symmetric monoidal category, 291

equivariant prefactorization algebra, see prefactorization algebra, equivariant expectation value, 23, 131, 140 expected value, 23

factorization algebra, 207, 210 Beilinson–Drinfeld, 11 category of, 214 computations, 243 from cosheaf, 225 differentiable, 213 equivariant, 241 extension from a basis, 232 homotopy, 212 from local Lie algebra, 230 in a multicategory, 212 in quantum field theory, 215 Kac–Moody, 78, 246 limitations from point of view of quantum field theory, 215 local-to-global axiom, 207 locally constant, 14, 221 on manifolds with geometric structure G, 216 multiplicative, 210 multiplicative homotopy, 213 pushforward, 232 relationship with En algebras, 221 with respect to a factorizing basis, 233 on a site, 216 weak equivalence of, 213 factorization envelope, 75, 188, 230 twisted, 77, 104 factorization homology, 220, 223 βγ system, 248 of Kac–Moody factorization algebra, 246 of quantum observables of free scalar field theory, 244 relationship with Hochschild homology, 223 of universal enveloping algebra, 243 factorizing basis, 233 field theory Abelian Chern–Simons, 115, 128, 249 Abelian Yang–Mills, 99 Atiyah–Segal, 121 chiral conformal, 148, 154 free, 98 free βγ system, see βγ system

Index

free scalar, 24, 27, 99 functorial, 121 interacting, 40 quantum, 24 topological, 118 translation invariance of free scalar, 132 vacuum for free scalar field theory, 140 free scalar field theory, see field theory, free scalar functorial field theory, 12, 220 Gaussian integral, 25 Gaussian measure, 25 Green’s function, 38, 101, 124 free scalar field, 140 Hamiltonian, 107, 131, 135 Hamiltonian formalism, 131 Hochschild cohomology for En algebras, 271 Hochschild homology relationship with factorization homology, 223 Hochschild–Kostant–Rosenberg (HKR) theorem, 271 holomorphically translation invariant prefactorization algebras, 149 homotopy colimit, 362 interacting field theory, see field theory, interacting interaction term, 41 Kac–Moody algebra, see Lie algebra, Kac–Moody Koszul duality, 250 Lie algebra, 292 acting via derivations on a prefactorization algebra, 83 Chevalley–Eilenberg complex, 294 cohomology, 293 coinvariants, 292 differential graded (dg), 295 enveloping algebra as prefactorization algebra, 59 Heisenberg, 92, 104 homology, 293

385

invariants, 292 Kac–Moody, 190 module over, 292 representation, 292 universal enveloping algebra of, 59, 293 line operator, 249, 262 linking number, 122, 128 local cochain complex, 226 local Lie algebra, 75 central extension, 77 local-to-global axiom, see factorization algebra, local-to-global axiom locally constant factorization algebra, see factorization algebra, locally constant measurement, 4 Moyal product, 127 multicategory, 289 map of, 290 symmetric monoidal envelope, 291 observable, 3 as cocycle in cochain complex, 9 support of, 95 Wilson loop, 122 observables classical, 34, 93 classical as h¯ → 0 limit of quantum, 35 quantum, 31 relationship between quantum and classical, 35, 124 shifted Poisson structure on classical, 102 structure maps as Moyal product on quantum, 127 operad, 283 BD, 287 P0 , 287 algebra over, 286 colored, 289 endomorphism, 286 operator product expansion, 148 ordinal category, 275 parametrix, 307 partition of unity, 300 path integral, 23 precosheaf, 298 prefactorization algebra, 45, 47, 49

386

Index

bimodule as a defect, 52 category of, 50 classical observables as, 93 derivation of, 83, 133 differentiable, 65 encoding standard quantum mechanics, 54 enveloping, see factorization envelope equipped with action of a Lie algebra, 83 equivariant, 79 examples from topology, 221 holomorphically translation-invariant, 149 holomorphically translation-invariant with a compatible S1 action, 157 locally constant, 51 morphism of, 49 motivating example of, 3 multicategory of, 50 on a site, 216 on manifolds with geometric structure G, 216 physical interpretation of, 9 pushforward, 112, 117 smoothly equivariant, 84 smoothly translation-invariant, 134 structure on observables, 31 translation-invariant, 132 unital, 46, 48 presheaf, 297 propagator, 38, 124 pushforward, see prefactorization algebra, pushforward, see factorization algebra, pushforward, 232 with respect to a factorizing basis, 233

quantization Batalin–Vilkovisky, see Batalin–Vilkovisky formalism canonical, 112 as enveloping algebra of Heisenberg Lie algebra, 92 quantum field theory, 24 comparison of formalizations, 11 Osterwalder–Schrader axioms, 14 Segal’s axioms, 12 Wightman axioms, 14 quantum group, 249

quantum mechanics, 106 measurement in, 4 as a prefactorization algebra, 54 as motivating prefactorization algebras, 3 Schwartz functions, 142 sheaf, 297 simplicial Abelian group, 279 simplicial bar construction, 368 simplicial set, 276 site of manifolds, 62, 311 smoothly equivariant prefactorization algebra, see prefactorization algebra, smoothly equivariant stalk, 317 state, 139 translation-invariant, 139 vacuum, 140 symmetric monoidal envelope, 291 Tannakian reconstruction, 250 tensor product bornological, 331 relationship between completed projective and convenient, 347 relationship between projective and bornological, 347 topological chiral homology, 223 topological field theory, 14, see field theory, topological translation invariance of free scalar field theory, 132 translation invariance vs. translation equivariance, 131 universal enveloping algebra, see also Lie algebra, universal enveloping algebra of vacuum, 140 free scalar field theory, 140 massive, 140 vector space bornological, see bornological vector space convenient, see convenient vector space locally convex topological, 325

Index

vertex algebra, 145, 146 βγ , 173 arising from factorization algebra, 157 arising from Lagrangian field theory, 148 field in, 145 Kac–Moody, 191 reconstruction theorem, 161 relationship with operator product expansion, 148 vertex operator, 147

Weiss cover examples, 209 Weiss topology, 209 Weyl algebra, 106 rigidity of, 108 Wick’s formula, 25, 39 Wick’s lemma, 25, 130 Wilson loop, see observable, Wilson loop Yang–Mills field theory, see field theory, Abelian Yang–Mills

387